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NASA Contractor Report 172342
I-
| Supersonic Second Order
| Analysis and Optimization
| Program User's Manual
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:.: W, C. Clever• _:,
ROCKWELL INTERNATIONAL CORPORATIONLos Angeles, California 90009
CONTRACT NASI-15820September, t984
{NAS_-C[_-1723q2) SUP£&SGNIC S_CGND ORDER
A_ALYSIS AND CP21_IZA_ICN P_CGBA_ USER'S
_NUAL [_ockwell Interr, ational Corp.) 223 pCSCL 0 IA
G3/02
N87-IC842 I L,
Uncla s
_3855
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National Aeronautics andSpace Administration
.LanoIoy Fllol_arch ContarHampton, Virginia 23665
https://ntrs.nasa.gov/search.jsp?R=19870001409 2020-07-01T15:03:39+00:00Z
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NASA Contractor Report 172342
SUPERSONIC SECOND ORDER ANALYSIS AND
OPTIMIZATION PROGRAM USER'S MANUAL
W.C. Clever
ROCKWELL INTERNATIONAL CORPORATION
Los Angeles, California 90009
CONTRACT NASI-IS820
SEPTEMBER, 1984
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FORWARD
This report was prepared by the Los Angeles Aerodynamics Group ofRockwell International, Los Angeles, California, for the Langley
Research Center, National Aeronautics and Space Administration, Hampton,
Virginia. The work was performed under Contract No. NA51-15820,
"Development of Second Order Potential Analysis/Design and Full
Potential Analysis Aero Prediction Technology for Hypersonic
Configuration Design." Mr. Noel Talcott and Mr. Kenneth M. Jones were
the Project Monitors of this contract.
Mr. E. Bonnet was the Program Manager and Dr. W. C. Clever was theprincipal investigator.
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SUMMARY
_pproximate nonlinear inviscid theoretical techniques for predictingaerodynamic characteristics and surface pressures for relatively slendervehicles at supersonic and moderate hypersonic speeds were developed.Emphasis was placed on approaches that would be responsive to conceptualdesign level of effort. Second order small disturbance and full potentialtheory was utilized to meet this objective.
Numerical codes vere developed for relatively general three-dimensionalgeometries to evaluate the capability of the approximate equations of motionconsidered. This report represents a user's manual for the second orderanalysis and optimization codes. Contained herein for each program is a briefdescription, its input data, variables, and subprograms, a floe diagram, and asample test case. Results from the computations indicate good agreement withexperimental results for a variety of wing, body, and wing-body shapes. Casecomputation times of a minute on a CDC 176 were achieved for practicalaircraft arrangements.
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TABLE OF CONTENTS
INTRODUCTION
ANALYSIS PROGRAM WBODY
DISCUSSION
INPUT DATA
SUBROUTINE DECRDI
LIST OF VARIABLES
SUBPROGRAMS
FLOW DIAGRAM
TEST CASE
OPTIMIZATION PROGRAM OPT
DISCUSSION
INPUT DATA
LIST OF VARIABLES
SUBPROGRAMS
FLOW DIAGRAM
TEST CASE
REFERENCES
Page
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19
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112
137
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144
148
217
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INTRODUCTI ON
An examination of the literature for supersonic/hypersonic aircraft
provides an indication of the flexiblilty and generality required for an
analysis technique. Typical configuration development variables include wing
section, incidence, height, dihedral, planform, effectiveness of longitudinal
control surfaces for trim, effectiveness of empennage for directional
stability, and propulsion system-airframe interactions.
Response to these requirements at the conceptual design level have beenby the hypersonic impact methods and the linearized analysis and designalgorithms. These approaches can treat complex geometries efficiently withminimum response time and cost. Shortcomings exist with both methods. Theimpact methods ignore component interference effects and crossflow
interactions. The linearized approaches exclude shocks, vorticityand entropywakes and layers.
A need exists for an improved analysis technique to optimize vehiclesdesigned to travel at supersonic/hypersonic speeds. The technique should bemore accurate than simple noninterfering panel methods and morecomputationally efficient than the current explicit finite-difference methods.Enough of the physics of the flow should be included to allow realisticoptimization and permit considerations of component interaction. Nonlinearpotential theoretical formulations hold the promise of meeting theseobjectives for preliminary vehicle definition efforts.
To satisfy the analysis needs, a program was undertaken to advance the
aerodynamic prediction capabilities at the conceptual design level. Numerical
second-order potential small disturbance analysis was developed as a first
step up from the widely used linear theory. Such formulation incorporates
nonlinear behavior by iteration of the Prandtl-Glauert approximation. Thisapproach is known to extend the prediction success for airfoil and cone
surface pressures to substantially higher values of the hypersonic similarity
parameter than the first-order theory. References 3 and 4 contain the details
of the theoretical development of the second order analysis and optimizationmethod.
This report provides a user's manual for the second order analysis andoptimization codes. Contained herein for each program is a brief description,its input data format, a complete list of variables and subprograms, flowdiagrams, and sample test cases.
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The computer program entitled "SOPA-Second Order Potential Analysis andOptimization Program" can be obtained for a fee from:
Computer Software Management andInformation Center (COSMIC)
112 Barrow Hall
University of GeorgiaAthens, GA 30602
(404) 542-3265
Request the program packgage by the designation LAR 13314. The programs arewritten in FORTRAN V for use on the Control Data 5600 and the Cyber series ofcomputers.
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ANALYSIS PROGRAM WBODY
DISCUSSION
Computer program WBODY performs a linearized analysis of wing bodycombinations using axis and surface singularities. The solution satisfies thePrandtl-Glauert equation with boundary conditions as prescribed by theassumptions of thin wing theory. A second order solution may then beperformed using the results of the first order (linearized) solution. Thesecond order solution on the lifting surfaces is performed by using amodification of the exact result available for floe in two dimensions. On the
body the second order solution combines an exact axisymmetric axial solutionwith the first order cross flow solution. The second order solution is notvalid for transonic Mach numbers and should not be used for Mach numbers
between 0.70 and 1.60. All output is placed in a dataset which may be usedfor computer graphics.
If a body is present and the use of axis singularities is indicated,isolated body axisymmetric axial and cross flow solutions will be performedfirst, using only the axis singularities. The axis singularities consist oflinearly or quadratically varying line sources and quadratically or cubiclyvarying line doublets. The singularity and control point spacing are input bythe program user. Either mass flux or velocity boundary conditions may bespecified. A second order axial line source solution will be performed ifindicated by the input.
The body panels are quadrilaterals of arbitrary shape having a constantsource distribution. The lifting surface panels are quadrilaterals having twostreamwise edges with a constant vortex distribution to simulate lift, and aconstant or chordwise and spanwise linearly varying source distribution tosimulate thickness.
The first order solution is performed by satisfying the condition of zeronormal velocity, or zero mass flux if desired, at the panel control points ofthe body and lifting surfaces. The lifting surface boundary conditions arelinearized and are satisfied on the mean camber line. The normal velocity ornormal mass flux is composed of the sum of contributions from the free stream,angle of attack, and perturbations from the axis singularities, body panels,
and lifting surface panels.
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The complete first order solution is composed of a linear combination offour basic solutions:
I. The cross flov or add load solution. This solution contains
no lifting surface thickness or camber.
. The camber solution. This solution satisfies the boundaryconditions for the lifting surface camber vith zero freestream velocity and zero normal velocity (or mass flux)on the body panels.
0 The lift due to thickness solution. This solution representsthe additional load due to the normal velocities induced
by the thickness of the lifting surfaces.
0 The lift due to the body solution. This solution representsthe additional load due to the normal velocities induced
by the body in axial flow with no camber.
The second order solution, if desired, is performed using the results ofthe first order (linear) solution.
INPUT DATA
Data is input using subroutine DECRD1, described on page 19, and isstored in the array called "DATA". All locations are initially setequal to 0. Each case contains:
Miscellaneous data
Body geometry dataLifting surface geometry
locations 6 - 88locations 701 - 1900
locations 1 - 5, 89 - 700and, 1201 - 1900
All of the data except the lifting surface geometry data must be input first,vith the last card having a negative location number. This data must include
any input for body panels and axial singularities.
The lifting surface geometry data follovs vith the last card for each liftingsurface having a negative location number.
The last surface or last card for a given case is indicated by a positivevalue in location 1.
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The body or suzface gecmetry may be read from an APAS (Aerodynamic Preliminary
Analysis System, references 5 and 6) file using data location 14.
The first card in each case may be a title card containing up to 72characters. If the first card contains a blank or a minus sign in the firstcolumn and blanks in columns 2 thru 4, it is assumed the card containsnumerical data, and is not a title card.
In addition the first card for each lifting surface may be a title cardcontaining up to 16 characters, i.e. cols 1 - 16. This title will be usedfor the name of the surface.
The following is a description of the possible input for each case.
L_ VARIABLE
I < O.
> O.
> 1.1
2 N
3 M
7
8 X.y
ACALC
DESCRIPTION
The run will terminate.Indicates the last surface has been read.
The run will terminate after the slender bodygeometry (if any) has been printed.
The number of panels in the chordwise direction forthis surface.
The number of panels in the spanwise direction forthis surface.
2. Compute the aerodynamic influence matricies andthe quasi-inverse matrix and store on unit ll.
1. Compute source influence matrix and store on unitll. The vortex and quasi-inverse matricies areassumed to already exist.
0. Compute new aerodynamic influence matrix.-1. Use aero influence matrix stored on unit 11.
-2. Read aero and quasi-inverse matricies from unit ll
(A second order solution always requires aquasi-inverse matrix).
If x.y < 0. a 2nd order solution is performed.If y > 6 d/dy of v terms are included.
5
10
12
13
14
ITYPE
CENT
the type of source panels used.If =-2. linearly varying source panels are used
(spanwise and chordwise without leadingor trailing edge panels).
If =-1. linearly varying source panels are used(chordwise varying only without leading
or trailing edge panels).If = 0. Constant source panels are usedIf = 1. linearly varying source panels are used
(chordwise varying only with leadingand trailing edge panels).
If = 2. linearly varying source panels are used(spanwise and chordwise with leading
and trailing edge panels).
If < 0.
If<-2.
Only the geometry and the axialsingularity solution will be calculated.
The program will stop after all of thegeometry (including axial singularitygeometry) is calculated.
1. Control point at panel centroid.Control point at panel center otherwise.
1.0 Obtain body and aero surface geometry from input
1.0 Obtain body geometry from APAS (panels)1.1 Obtain slender body geometry from APAS,
create panels, and save after aero geometry.1.2 Obtain slender body geometry from APAS, modify,
create panels, and save after aero geometry.
2.0 Same as 1.0, 1.1, and, 1.2, but includes
2.1 aero surface panels and geometry.2.2
2.0 Use twist and cambers from APAS and input data.2.1 Use twist and cambers from input data only.2.2 Use twist and cambers from APAS dataset only.
3.0 Obtain only aero surface panels and geometry fromAPAS (no body).
3.0 Use twist and cambers from APAS and input data.
3.1 Use twist and cambers from input data only.
3.2 Use twist and cambers from APAS dataset only.
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1. Print source dz/dx and z/c matricies
2. Print both source and camber arrays
3. Print camber slope matrix
IPRNT 0. No printing of panel geometry.
1. Print body panel geometry (source panels).
2. Print vortex panel geometry
3. Print body panel and vortex panel geometry
-I. Print body panel geometry from APAS dataset.
-2. Print vortex panel geometry from APAS dataset.
-3. Print body panel and vortex panel geometry (APAS)
0.K Z > 0 Print vn due to thickness.
K > I Print u due to thickness.
K > 2 Print v due to thickness.
Z > 3 Print w due to thickness.
K > 4 Print phi due to thickness.
INTMED Various orders of intermediate printout (-1. to 4.)
-1. Least printout (no upper and lower surface Cp's)2. Hore Printout
Prints add load solution and v-normal
(w-w), (w-b), (b-w), (b-b)3. Above +
4. Above + 2nd order boundary condition solutions
5. Above + odd and even 2nd order velocities and Cp's.
DPLOT .
1.
2.
> 2.
No data is placed in a plot dataset.Data is placed in an APA5 dataset (geometry)
Data is placed in a UDP dataset (for plotting).
Data is placed in a UDP dataset (extended)
DPLOT > 2. is required if the dataset is to be used fora 2nd order optimization calculation.
XIJ.XKL Detailed influence matrix calculation printout for
vortex panels. From subroutine VORTEX.
XIJ = 3 digit number for control point
XKL = 3 digit number for panel number
XIJ.XICL Detailed influence matrix calculation printout forsource (thickness) panels. From subroutine PHISll.
XIJ = 3 digit number for control point
XKL = 3 digit number for panel number
23 APRNT = IJ.KKK
.ne. 0. Print the aero influence matrix (vortex and body)
.gt. 0. Only normal velocities are printed.
.it. 0. All velocities and phi are printed.
I or J = 0 nothingI or J = I vortex
I or O = 2 bodyI or O = 3 both
I - influenced
J - influencing
ll.KKK vortex on vortex only
12.KKK body on vortex only21.KKK vortex on body only
_BS(APRNT) = 22.KKK body on body only13.KKK all on vortex
31.KKK vortex on all
23.KKK all on body
32.KKK body on all32.KKK all on all
The influence matrix printout will stop after KKK rows arePrinted. If KKK = 000 all of the rows are printed.
24 SPRNT = IJ.KKK
.he. 0. Print the source (thickness) influence matrix
> 0. Only normal velocities are printed.< 0. All velocities and phi are printed.
I = 0 nothingI = 1 vortex
I = 2 bodyI = 3 both
I - influenced panel
IJ.KKK
ABS(APRNT) = 2J.KKK3J.K_
source on vortex onlysource on body onlysource on all
The influence matrix printout will stop after KKK rows are
printed. If KKK = 000 all of the rows are printed.
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25 > -1.
25-28
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Print perturbation velocities from axial singularities
= 0. Print velocities from sources and doublets= 1. Print velocities from sources= 3. Print velocities from sources (lst and 2nd) and doublets= 4. Print velocities from sources (lst and 2nd)> 4. Print velocities from sources (lst and 2nd) and doublets
Surface to be extended to centerline.
the surface will only be extended when the aeroinfluence matrix is calculated.
S = surface number (in ascending order).
K = 0 Extend inboard panels with zero sweep.K = 1 Extend inboard panels vith same sweep.
Z = 2 Extend inboard panels with negative sweep.
i.e. if DATA(26) = 2.0 the calculation of the aero
influence matrix for surface # 2 ,i11 be
performed by extending the inboard panels to
y = 0. with zero sweep.
the actual panel will be extended and locations 711-714should be used.
beta * y / x is printed for each panel
Mach number used for order 2 solution CK2 array.
MS > 0. use MS as Mach number.
MN = 0. use freestream Hach number (generally used).HN < 0. use normal Mach number.
if < 0. the maximum value of CK2 = - HN
the x value of the pivot point for angle of attack.
used for second order theory only.
For computing On, Cr, Cw, Cy
For non-slnmetric rolling moment only
For non-symmetric rolling moment only
S.K
If>0
if<0
X00
XCG
YCG
ZCG
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3839
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MACH
RD
RS
CBAR
CAVG
SREF
SPAN
RATIO
The Mach number
The x/c fraction of each panel where the normalvelocity is interpolated to calculate the zerosuction drag. If 0. a value of 0.515 is used.
The x/c fraction of each panel which is used tocurvefit (Cp,x) in order to obtain an approximatevalue for the leading edge suction. (default = 0.250)
Drag Polar (41 points calculated)
Delta CL for drag polar. Default = .05
Starting CL for drag polar. Default ffi0.
The reference chord length for computing Cmif 0. CBAR = CAVG is used.
The reference chord length. If 0.CAVG = SREF/SPAN
The reference area. if 0.SREF = total area
Span, any consistent set of units may be used. Thisvalue is used for reference purposes only. If 0.SPAN = 2. * Y-max
The chordeise control point location, if 0. default is0.875 for Mach> 1.0.875 for Mach< 1.
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45
Data locations 45 - 47 apply to 2nd order solutions only.
IJK Printing of odd and even symmetry u,v,w velocities.
if DATA(45) = LMN. I = L, J = M, K = N
! J K
= 0 No velocities printed ffi 0 none = 0 none= 1 u velocities printed ffi 1 odd = 1 add load= 2 v velocities printed = 2 even = 2 twist and camber> 2 All velocities printed • 2 all = 3 thickness
= 4 axial>4
e.g.
use DATA(45) ffi231. I ffi2, J = 3, K = I
To print odd and even symmetry thickness v velocities.
46 IJK
47 IJK
48 J
Printing of odd and even symmetry d/dx of u,v,w
Printing of odd and even symmetry d/dy of v
I > 0 is required
Printing of d/dx of camber and thickness normalvelocities.
J = 1 Camber only.J = 2 Thickness only.J • 2 Both
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50 ALPHA0 The angle of attack of the surfaces with respect tothe x,y plane (degrees). Used for second order
theory only. ALPHAD is made up of ALPHA0 and
the angle of attack of the freestream with respectto the x-axis, ALPHAI.
i.e. ALPHAI = ALPHAD - ALPHA0
For a first order solution only ALPHAD matters.
51 ALPHAD(1) The angle of attack of the configuration with respectto the freestream (degrees).
52 ALPHAD(2)58 ALPHAD(8) (maximum number of angles of attack = 8)
If ALPHAD(K) > 90. an add load solution is perfomed.
i.e. (ALPHAD = 1.0) - (ALPHAD = 0.0)
61 CAMTHK(1) Input in form OB-OV.CAM-THK, and used with ALPHAD(1)
OB = The order of the Cp on the body (one digit).OV = The order of the Cp on the lifting surfaces)CAM > I Camber included.
> 1 Thickness included.CAH = 0 No Camber included.THK = 0 No thickness included.
0B, OV, CAM, THK are each one digit.
If = O. OB = 4, is used for the body Cp,
Or, is determined by DATA(8), andcamber and thickness are included
IfOV = I, unless DATA(8) < 0.
OV > 2, OV = 2 will be used, and also thedifference between the Ist and 2nd
order delta-Cp's will be printed.
If < 0. Same as above except the 2nd order axialsolution (if performed) is used tocalculate u,v,v.
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in
OB Cp
The body pressure formula isdetermined by the value of OB.
12345
Cp = - 2 * U
Cp = - 2 * U - beta2 * u*u - v*v - w*w
Cp = - 2 * u - u*u - v*v - w*wCp = Isentropic pressure formula
Cp = Isentropic for alpha = 0. + isentropic add load
u,v,v use 1st order axial contributions if CAMTHK > 0u,v,w use 2nd order axial contributions if CAMTHK< 0
e.g. CAMTHK = - 31.02
- indicates, 2nd order u,v,w from axial solution (if performed).3 indicates, pressure formula #3 on body.I indicates, 1st order Cp on lifting surfaces.0 indicates, camber is not included.2 indicates, thickness is included.
62 CAMTHK(2) used with ALPHAD(2)
68 CAMTHK(8) used with ALPHAD(8)
The following data (89-700) are read for each lifting surface.
89 TC
9091929394
The t/c due to thickness for this lifting surface
If < 0. a thickness influence matrix is calculated
although TC = 0 is used. A second order solutionwill always calculate a thickness influence matrix.
Locations 90-94 are the thickness coefficients
If 90-94 are all 0. a NACA 4 digit airfoil is used.If CO < 0. a biconvex airfoil is used.
COC1C2C3C4
The coefficient of the SORT(x) term for thicknessThe coefficient of the x term for thicknessThe coefficient of the x**2 term for thicknessThe coefficient of the x**3 term for thicknessThe coefficient of the x**4 term for thickness
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102103
104"105
106107
108109110
111
112113
SWEEPL
SWEEPTROOT
ROOTX0
R00TZ0
SPANOXSP
FLAP
DIHDRL
RATI 0Y
SYM
ND
MD
The leading edge sweep in degrees. This value is
ignored if 103 is .It. 0., which means a planform
shape is to be read in.
The trailing edge sweep in degrees
Root dimension or chord length along the symmetry axis<0. the planform shape is read in from 241-320
>0. this value is used to calculate the geometryThe x value at the start of the root.
The z value for the root. This value will be used
with 109 or will be added to values from 121-160.
The value of the SPAN. Used with even spacing.
If < 0. the chordwise panel spacing is evenIf = 0. the chordwise panel spacing is cosineIf > 0. the chordwise panel spacing is half cosineThe x/c for the flap hingeThe dihedral angle in degrees. (used with 105)The spanwise control point location. If 0., the
centroid is used unless location 201 is nonzero.
= 0. Symmetry about y=0. is assumed0. No symmetry
The number of chordwise locations of camber input.The number of spanwise locations of camber input.
if MD < 3 only the first value will be used.i.e. all span stations will have the same camber.
121161
201 Yc(J)
Planform Shape
z values at the y coordinates
y coordinates. (if ((Y(2)-Y(1))**2+(Z(2)-Z(1))**2) is
0. The semi span is broken into M equal segments)
there must be M+I values input
y coordinates for the control points. Nonzero valueswill be used to override values based on location 110.
241
281
_z(J)
rrz(j)
The leading edge coordinates at Y(J)
These values are considered only if 103 is < 0XLE(1) Corresponds to the coordinate on the axis
XLE(m+I) corresponds to the coordinate at the tip
any values which are exactly O. will be changed
to make the edge straight between the twosurrounding nonzero values.
The trailing edge coordinates. Same format as 241-280
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321341361381
401
702703704
7O5
706
707
708
XD(I) The x/c values where the camber is input. See 112YD(J) The y values where the camber is input. See 113ZD(J) The z values where the camber is input.
TWIST(J) The twist in degrees at the above values of (y,z)
The values of dz/dx for the camber.401 to 400+ND are for YD(1) at XD(1) to XD(ND)401+ND to 400+ND*2 are for YD(2) at XD(1) to XD(ND)
These values are interpolated to obtain the boundaryconditions at the control points.
Body Geometry
NYNXNX
NC
IJ.K
ECC
The number of panels around the body (half).The number of panels in the • direction on the body.The number of singularity segments on the slender
body. If < 0 the singularity segments rill beshifted along the Mach lines (supersonic only)
The number of control points on-the slender body.
I = the order of the source singularities (1,2).J = the order of the doublet singularities (2,3).
If I = 0 I = 1 is used.If I > 2 I = 2 is used.
If J = 0If O > 3
J = 2 is used.J = 3 is used.
There is always a 1st order line source at the nose.There is always a 2nd order line doublet at the nose.
IJ < 0 exact conical solution desired at nose.
IJ > 30 the nose 1st order line source strength = O.the nose 2nd order line doublet strength = O.
K > 0 a second order axial solution is performed.
K=I
K=2no r*phir**3 term included in order 2 solution.
r*phir**3 term included in order 2 solution.
The eccentricity of the body cross sections.
Area = pi * r(1)**2 for body cross sections.
Velocity boundary conditions on body (beta2x = 1.0)Mass flux boundary conditions on body. (beta2x = beta2)
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709
710
711
712
713
714
715
721722
723724
725726
M.N
IU
_i(1)WBI(2)_I(3)WaX(4)
>0M>I
>2
Print source and doublet solutions.
Print shifted singularity points and loads.Print source and doublet coefficients.
=I
N=2
>2
Print source influence matrixPrint doublet influence matrixPrint both influence matricies
unit number for placing axial loads in a plot dataset(use unit 12)
= O. the body and Ist surface intersection is computed.
= O. the body and 2nd surface intersection is computed.
= O. the body and 3rd surface intersection is computed.= O. the body and 4th surface intersection is computed.
DELTA Used to calculate the maximum allowable slope of
bodies. For the axial singularity calculation a
conical nose extension is constructed tangent to thebody such that:
dr/dx = 1.0 / beta - DELTA if delta > 0.
This operation is performed only for Mach > 1.0 .
on the region of the actual body where,
dr/dx > 1.0 / beta - DELTA
Tangent cone formulas are used to calculate Cp's.
The type of pressure coefficient calculation, and theangle of attack, for 721-726, are determined fromlocations 51-58 and 61-68.
THETA- 1
THETA - 2
THETA - 3
THETA - 4
THETA - 5
THETA - 6
the Ist theta for Cp computation on body.
the 2nd theta for Cp computation on body.
the 3rd theta for Cp computation on body.
the 4th theta for Cp computation on body.
the 5th theta for Cp computation on body.
the 6th theta for Cp computation on body.
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751
801
851
901
10011101
1201
1801
VSHeLL(ISF)= 1. the surface is a shell composed of vortex panels
only the upper surface is considered.= O. the surface is a normal lifting surface.
XINLET(I) If .he. 0. The • station is assumed to have an inletthe mass floe is (I.-XINLET(I)).
XG(I) The • coordinates of the body geometry sections.
zz(I) The • coordinates of the body panel cross sections.if XX(I) = 0., and DATA(703) > 0., XX(I) = XG(I)is assumed.
RR(I) The r values of the body geometry sections.
x(I)xc(I)
The • coordinates of the slender body singularities.
The • coordinates of the slender body control points.
CAM(IJ) The camber for each panel. Input as dz/dz at the
control point (each surface input separately),
unless the input is being read through an APAS
dataset. For an APAS input all camber valuesare input at one time and the value of DATA(14)must be considered.
i.e. If the data is not being input through an APA$dataset, DATA(1200+I) is the initial camber dz/dx
for the I'th panel of the surface being input.
TWIST(J) The twist for each span station. This is for APAS
dataset inputs only, and the twist values for all
span stations are input at one time. See DATA(14)
for additional information on twist input.
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19961997199819992000
FLOW(l)FLOW(2)FLOW(3)FLOW(4)FLOW(5)
FLOW(1)
Mass flow coefficient for inlet # 1
Mass flow coefficient for inlet # 2Mass flow coefficient for inlet | 3Mass flow coefficient for inlet | 4Mass flow coefficient for inlet | 5
= Average value of (1.-beta2x*u) for all field pointof inlet I (see location 708).
maximum of 200 field points
200120062011201620212026
x, y, z, inlet |x, y, z, inlet |x, y, z, inlet |x, y, z, inlet |x, y, z, inlet #x, y, z, inlet |
for field point |1for field point |2for field point |3for field point |4for field point |5for field point |6
inlet # = 0 means the point is not an inlet point.
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SUBROUTINE DECRDI
Subroutine DECRD1 is used to read input data from a fixed block dataset.The input data which is read is placed in floating point format into locationsof the array "DATA" which appears in the argument list. The subroutine willalso read a single title card of up to 72 characters and place the literaldata in the array "TITLE" which also appears in the argument list. SubroutineDECRD1 allows input in two different formats, called decrd format and freeformat. A description of these input formats will follow.
The first time DECRDI is entered, the first record is assumed to be in
decrd format with LRECL = 72. LRECL is the length of each record which isread (maximum LRECL = 132).
If the characters "C","D", or "F" appear in column I, the card will notbe read for any data (except for LRECL).
A 'C' indicates a comment card.
A 'D' indicates the following cards are to be read using decrd format.An 'F' indicates the following cards are to be read using free format.
The value of LRECL appears in the card with an F in column I in the formLRECL(KLM), where KLM is a three digit integer. It will remain constant each
time DECRDI is entered, unless it is reset.
Unless the first card containing data in each entry to DECRDI has a blank
or a minus sign appearing in column 1, with blanks in columns 2-4, the first
card is assumed to be a title card containing 72 characters of literal data.However, cards with the characters "C", "D", or "F" in column I do not count
as data cards and may appear before the title card. When using free format,
the first data card of an entry to DECRDI should be checked carefully to seethat the first four columns are of the correct form, to avoid confusion
between a card containing title data and a card with numerical data.
DECRD FORMAT
In Decrd format each data card has the format I12,5E12.5. The first
number on each card is an integer giving a location in the input array; the
following numbers on the card specify the values to be input into that and the
four immediately succeeding locations. These numbers can be input either in F
format, which must include a decimal point or in E12.5 format. Locations left
blank will remain unchanged. The last card to be read for each call to DECRDIis signaled by having a negative location number.
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FREEFORMAT
Data may be written in I, F, E, or D format separated by blanks, commas,
or semicolons. The data will be placed in consecutive locations of the array
"DATA" in the program. Data locations may be skipped by writing an X or using
consecutive commas with nothing other than blanks between. When data
locations are skipped, the previous values remain unchanged.
If the first piece of data on a card is a positive or negative integer,which is not a multiplying factor to be described below, it will be assumed tobe the location number of the next piece of data on the card. If this integeris negative this card will be the last card read until subroutine DECRD1 is
called again. This means the last card must have a negative location numberas the first piece of data on the card. If the piece of data is not aninteger, or if it is a multiplying factor, the data will be placed inconsecutive locations continuing from the previous card.
A semicolon can be used to designate the end of one card and the start ofa new card. This means that the next piece of data will be treated as if it
were the first piece of data on a new card. For this new card everything inthe previous paragraph applies.
An integer followed by '=' will cause the subsequent data to be placed inconsecutive locations beginning with the integer. Blanks may be placed beforeor after the '='.
A * preceeded by an integer is used to designate a multiplying factor,and will cause subsequent data to be repeated the integer number of times.E.g. 5 * 4.4 will result in the value 4.4 being placed in 5 consecutive datalocations starting with the appropriate data location. Writing 10*X willresult in l0 data locations being skipped. Spaces before and after a * areignored. A multiplier must be an integer (I format), and it must be > 1.
Everything within ( ) will be ignored, unless it is preceeded by amultiplying factor which is greater than one. E.g. 5*( 3.2 4.6 x 5 ) will
cause the sequence of data ( 3.2 4.6 x 5 ) to be repeated 5 times. Nestedparentheses are not allowed.
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LIST OF VARIABLES
GENERAL VARIABLES
CHORD(J)
IJS(IS)
IJ0(J)
ISO(ISF)
JS(IS)
MS(IS)
NB
NCHORD(J)
NS
NSF
NSPAN(ISF)
NSL
NST
NSTS
NTB
NTL
NTPNTS
NTSL
NTV
The chord value at the centroid of span station JThe value of IJ where section IS beginsThe number of the first panel at span station JThe section number where surface ISF begins.The span station where section IS beginsThe number of span stations in section ISThe number of basic solutions.
The number of panels at span station JThe total number of sectionsThe total number of surfaces
The number of span stations on surface ISFThe number of vortex shell sections
The total number of span stations on all liftingsurfaces.
The number of source span stationsThe number of body panels.The number of vortex shell panels.The total number of panels.The number of source parameters.The total number of vortex shell span stationsThe number of vortex panels.
LIFTING SURFACE PANELS
CAM(IJ)
COSZ(IJ)
cx(z,zs)DS(J)
DWDX(IJ)
DX(I,IS}ETA(J)
PA(IJ)SINZ(IJ)TWIST(J)
WTI-TK(IJ)X(KL,IC)XC(IJ,1)XC(IJ,2)
XLE(Z,J)XLE(2,J)XM(I,IS)rm(z,j)
The normal velocity at the control point due to
The cosine of the normal of panel IJ
x/c values for the control points of section IS
Width of span station JThe local dw/dx at thickness control point IJ
Delta (x/c) for the panels of section IS
Fraction of span running distance for (YCC(J),ZCC(J))c_ber.
The area of panel IJThe sine of the normal of panel IJ
Twist of span station JThe local dz/dx at thickness control point IJ
x values of the corners of vortex panel KL (4)
x value of the centroid for vortex panel IJx value of the control point for vortex panel IJ• value at the leading left edge of span station J.• value at the leading right edge of span station J.x/c values for the midpoints of panels of section IS• value at the trailing left edge of span station J.
21
rrs(2,J)xx(i,Is)Y(KL,IC)Yc(iJ)Ycc(J)Yo(z,a)YO(2,J)Z(KL,IC)ZC(IJ)zcc(J)ZTHK(IJ)
ZO(1,J)zo(2,o)
x value at the trailing right edge of span station O.x/c values for the panels of section IS
y values of the corners of vortex panel KL (2)
y value of the control point for vortex panel IJ
y value at the control point of span station J
y value at the left edge of span station J.
y value at the right edge of span station J.
z values of the corners of vortex panel KL (2)
z value of the control point for vortex panel IJ
z value at the control point of span station J
The local z/c at thickness control point IJ
z value at the left edge of span station J.
z value at the right edge of span station J.
BODY PANELS
B(1) = SQRT(DY(1)**2 + BETA2*DX(I)**2) I = 1,4
BT(IC,IJ) tans**2 + beta2 for each side of body panel IJ
= I.- Mach**2 mass flux boundary conditions.
i.e. (1. + beta2x * u ) * XN + v * YN + ( w + alpha ) * ZN = 0.
BETA2X
IBB(K)
IJB(K)NX(K)NY(K)
= 1. velocity boundary conditions.The body station number of the first station in body
section K.
The panel number of the first body panel in section K.The number of • body stations for body section KThe number of panels at each body station (half) for
body section K.
XB(IC,KL) x values of the corners of body panel KL (4)
XINLET(IJ) mass floe coefficient of body panel IJ; 0. = impermeableXN(I,IJ) = • - component of body panel IJ normalXN(2,1J) = y - component of body panel IJ normalXN(3,1J) = z - component of body panel IJ normal
x.(4) = xs(z) / XN(8)XN(5) = XN(2) / SQRT(XN(2)**2 + XB(3)**2)
XN(6) = _(3} / SQRT(XN(2)**2 + X/q(3)**2)
X/q(7) = SQRT(XN(2)**2 + ZN(3)**2) / ]N(8)XN(8) = SQRT(beLa2*XN(1)**2 + XN(2)**2 + XN(3)**2)
XN(9) = SQRT( XN(1)**2 + XN(2)**2 + XN(3)**2) = 2. * area
XP(IC,KL) • values of the corners of body panel KL (4) planar
XB0(I) The • value of the center of body station I.
X0(1,IJ) • value of the centroid of body panel IJ
X0(2,IJ) y value of the centroid of body panel IJ
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XO(3,IJ) z value of the centroid of body panel IJYB(IC,KL) • values of the corners of body panel K_ (4)YP(IC,KL) • values of the corners of body panel KL (4) planarZB(IC,KL) • values of the corners of body panel FJ_ (4)ZP(IC,KL) • values of the corners of body panel KL (4) planar
COMPUTED VARIABLES
Cp(IJ,K) The delta-Cp across panel IJ from basic solution K.
UK(IJ,K)
vz(iJ,Z)WK(IJ,K)
PK(IJ,K)
The control point upper surface • velocity.The control point upper surface binormal velocity.The control point upper surface normal velocity.The control point upper surface velocity potential.
US(IJ)VS(IJ)WS(IJ)PS(IJ)
The thickness induced upper surface • velocity.The thickness induced upper surface binormal velocity.The thickness induced upper surface normal velocity.The thickness induced upper surface velocity potential.
The following variables are required for 2nd order solutions only.
WSX(IJ) d/dx of WS(IJ) source normal velocity. -WCX(IJ) d/dx of WK(IJ,2) camber normal velocity.
KKX The number of 2nd order boundary condition solutions.
= 6 if there is no body.= 9 if there is a body.
UBE(IJ,K) The even symmetry u velocities due to 2nd order b.c.
UBO(IJ,K) The odd symmetry u velocities due to 2nd order b.c.
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SUBPROGRAMS
MAIN
ABMAIN
ACMAIN
AERO2
AERO2B
AERO2P
AERO2V
AMINMX
APAS
APASB
AXXABA
AXXAIJ
AXXBDY
AXXBXR
AXXCSD
AXXLINAXXLOD
AXXNCP
AXXNF0AXXNF!AXXNOS
AXXSLXAXXTCNAXXUVMAXXVEL
AXXXIS
BDYAIJBDYSCE
FUNCTION
Sets the main array size for the program.
Reads input data, computes geometry, and sets arraysizes.
Controls the flow of the program.Controls the program flow for the computation of 1st
or 2rid order pressures and loads.
Computes the u velocities due to 2rid order boundaryConditions.
Computes the 1st or 2nd order pressures from theinduced velocities.
Computes the odd and even symmetry velocities inducedby the basic solutions.
Finds the largest, smallest or largest absolute valueof the elements of an array.
Reads geometry from APAS output.Modifies APAS body geometry coordinates to form desired
panel coordinates.
Checks if an axis singularity solution is desired.Computes axis singularity source and doublet
influence coefficients.
Computes body geometry for axis singularity solution.Prints axisymetric body geometry and coefficients of
source and doublet coefficients.
Computes coefficients of semi-infinite source anddoublet line singularities.
Computes x and r for axisymmetric body geometry.Computes Cp's and forces on isolated body from axis
singularity solution.
Integrates forces on body nose using Cp's from atangent cone solution.
Function for body nose curve fit.Function for body nose curve fit.Finds the point ehere a conical nose extension is
added to the body in order to avoid body slopes inexcess of the Mach angle.
Controls program floe for axis singularity solution.Computes tangent cone Cp's on the body.Computes velocities from axis singularities.Computes velocities at panel control points from
axis singularities.Computes influence coefficients for source and doublet
semi-infinite line singularities.Computes influence coefficients for body panels.Calculates u,v,e,phi influence coefficients for a
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BODYDS
BODYGBODYGMBODYRDBODYW
CLCM
DECRD1DISPLY
DMXMVEDSSPLY
ENDREC
FIELD
FXDX3
GEOM
GOMTRY
HSHLDR
INLET
INTRP3
INTRP4
INTRPX
INTRPY
INTSCT
LINEAR
MATRIX
MATRXF
MATRXT
MTXADD
MTXMLT
MTXMVE
unit strength source panel.
Prints arrays of characteristics at body panel control
points.
Computes body panel geometric characteristics.
Computes and prints body geometry from input data.
Reads body panel geometry from APAS dataset.
Finds the intersection of the body and the aerodynmaicsurfaces.
Computes the lift drag and moment characteristics of
aerodynamic surfaces.
Reads the input data.
Prints arrays of characteristics at the panel controlpoints of the aerodynmaic surfaces.
Moves the elements of a double precision array.Prints arrays of characteristics at the panel corner
points of the aerodynamic surfaces.Used with errset to check for APAS type influence
matrix.
Computes first order field point properties using
previously calculated influence coefficients andthe first order solution.
Integrates data using a third order curve fit through
the nearest four points.
Computes the geometric characteristics of the
aerodynamic surfaces and panels.
Computes the panel geometry for the aerodynamicsurfaces.
Solves sets of linear simultaneous equations in a
least square sense.
Performs solution while satisfying boundary conditions
at a set of inlet points.
Interpolates or differentiates data using a third order
curve fit through the nearest four points.
Interpolates or differentiates data using a third or
fourth order curve fit through the nearest four points.Interpolates or differentiates properties chordwise on an
aerodynamic surface using subroutine INTRP4.Interpolates or differentiates properties spanwise on an
aerodynamic surface using subroutine INTRP4.Finds the intersection of two lines.
Fills in aerodynamic surface geometry not input, and
Calculates geometry.
Computes aerodynamic influence coefficients for
source and vortex panels on aerodynamic surfaces.
Displays arrays of data.
Displays arrays of data.
Adds multiples of two arrays.
Multiplies two arrays.
Moves the elements of an array.
25
NOTZRO
ORTHO
PHISll
PHIZ_
PLOT
POLAR
QUADPL
SQFIT
THI CK
VORTEX
VTXAI J
VTXDRG
VTXDR2
VTXLFTWBINF
WBSOLWBUVWWBVEL
WINTRP
file
5689
101112
13
Checks to see if an array has any nonzero elements.
Solves sets of linear simultaneous equations using themethod of succesive orthogonalizations. A
quasi-inverse matrix may be computed or, if
previously computed, used to perform the solution.Calculates u,v,w,phi influence coefficients for aero
surface source panel edges on control points.
Calculates v (binormal) velocities on aerodynamicsurfaces from phi (for 2nd order solution).
Writes geometry and aerodynamic data on a disk unit
for computer graphics.
Calculates first order drag polar and aerodynamiccofficients.
Computes body source panel geometric characteristics.
Calculates leading edge suction using a least squareCurve fit of Cp to cot(x/c).
Computes wing thickness z/c, dz/dx, d2z/dx2 from input.Calculates u,v,w,phi influence coefficients for aero
surface vortex panel edges on control points.
Controls program flow for aerodynamic influenceCoefficient calculation.
Calculates vortex drag in the Trefftz plane.Calculates coefficients for VTXDRG.
Calculates vortex lift from leading edge suction.Prints source or vortex influence matricies for
aerodynamic surfaces.
Controls the matrix solution using subroutine ortho.Calculates boundary conditions for solution.
Calculates u,v,w,phi at panel control points.
Interplolates velocities to various points on panels.
use
input dataPrint outputscratch filescratch filescratch file
storage of influence matricies and quasi-inverseplot output fileAPAS geometry input file
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FLOW DIAGRAM
<==>* LINEAR ***********
V
<==>* GOMTRT ***********
V
**********
V
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V********** **********
* ACMAIN *<==>I<==>* GEOM *********** I **********
V**********
<==>* BODYG *<==>**********
V
**********
<==>* BODYRD ***********
V
I **********
<==>* AXXLIN ***********
V
V
V
**********
**********
<==>* AXXABA *<==> see Slender Body Flow diagram**********
V
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**********
<==>* AER02 *<==>I**********
V
IV
IV
IV
IV
**********
<==>* PHIXY ***********
V**********
<==>* AERO2V ***********
V• ********* **********
<==>* AERO2B *<=====>* WBSOL ***********
V**********
:<==>* AERO2P ***********
V
• ********* 1
V
**********
**********
I**********
* ORTHO ***********
**********
<=>* SQFIT ***********
V
**********<=>* VT_n'.FT ***********
V
**********
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SLENDER BODY CALCULATION
**********
* AXXABA ***********
IV
**********
* AXXBDY *<==>**********
**********
<==>* AXXLIN *
**********
********** t **********<==>* AXXSLX *<==>I<==>* AXXCSD *
• ********* [ **********
V V
V
IV
IV
lV
IV
V
I ********** **********
<==>* AXXAIJ *<=====>* AXXXIS *• ********* **********
V
V
**********
<==>* AXXLOD ***********
I<::>:*_.':<.......>:';_I:':<:>********** **********
**********
<=>* AXXXIS ***********
**********
<=>* AXXCOR ***********
31
I
TEST CASE
Results for the seventy degree delta wing-body arrangement of figure 1are presented in this section.
The aerodynamic paneling representation used in the analysis is presentedon figure 2 and is typical of the program geometrical graphics output.
Comparison of first and second order results with experiments for a Machnumber of 6 and an angle of attack of 8 degrees are presented, on figure 3.Improved wing surface pressure coefficient predictions are systematicallyobtained for the second order analysis with the exception of the root sectionon the compression side and are in reasonably good agreement withmeasurements. Additional results are presented in references 2 and 3.
Test case input is presented on pages 41 through 43, and detailed programprinted output is presented on pages 44 through 98. Typical aerodynamic datagraphics output is presented on pages 99 through 109.
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OF POOR QUALF_Y.
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TEST CASE INPUT DATA
WBODY.DATA(M600) 1ST-AXC
C
C
C
C
C
CCC
C
C
C
CCCCC
C
CCCCC
(I0 X II) (8 X II) VEL B.C. MEG
CALCULATE NEW AERO MATRIX AND INVERSE AND SAVE (=2.0)37X37
72.IF < 0. CALCULATES A SECOND ORDER SOLUTION.
8-1.
= 2.0 FOR SPANWISE AND CHORDWISE LINEARLY VARING SOURCE PANELS.10 2.
GEOMETRY CALCULATION ONLY12 0.
CONTROL POINT AT PANEL CENTROID (VORTEX PANELS)13 1.
NO PRINTING OF SOURCE (THICKNESS) DZDX AND Z/C MATRICIES15 O.
= 3.0 PRINTS ALL PERTURBATION VELOCITIES DUE TO THICKNESS18 3.5
PRINTS ALMOST EVERYTHING (EACH LOWER NUMBER PRINTS LESS)19 O.
PLOT DATASET (EXTENDED) NO PLOT DATASET (= 0. )20 3.
EXTENDS FIRST SURFACE TO PANEL CENTERLINE WITH NEGATIVE SHEEP26 1.2
MACH NUMBER35 6.0
PRINTS ODD AND EVEN SYMMETRY VELOCITIES45 334.
46 334.47 334.
PRINTS CAMBER AND THICKNESS NORMAL VELOCITIES48 0.0
ANGLE OF ATTACK ( > 90. FOR ADD LOAD )5I 99.0 99.0 8.0 8.0
IST ORDER AXIAL SOLUTION ( > 0. )
ISENTROPIC = 4 (ON BODY)
IST ORDER = 1 (ON LIFTING SURFACES)NO CAMBER = 0
NO THICKNESS = 0
61 41.00 42.00 41.00 42.00
41
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ORIGINAL PAGE 19
OF POOR QUALITY
NUMBER OF PANELS AROUND THE BODY (HALF)702 8.
AXIAL SINGULARITIES.
< 0 MEANS EXACT CONICAL SOLOTION AT NOSE.2 = AXIAL VARIATION ORDER OF SOURCE SINGULARITIES
3 = AXIAL VARIATION ORDER OF DOUBLET SINGULARITIES706-23.0
VEL B.C. = 0.708 0.
PRINT SOURCE AND DOUBLET SOLUTIONS.709 O.710 0.0
IF > 0. NO BODY AND SURFACE INTERSECTION IS COMPUTED
711 1. 1. 1. 1. 1.MAXI MUM ALLOWABLE SLOPE ON BODY = I. / BETA - DATA (715)
715 0.0029
PRESSURE COEFFICIENT CALCULATION ON BODY (AXIAL SOLUTION)720 4.0
ANGLE FOR PRESSURE COEFFI CIENT CALCIYLATION.
721 0. 22.5 45.X VALUES OF THE BODY GEOMETRY SECTIONS.
67.5 90.
801 0.0 1.00 1.09 1.10 1.20806 1.30 1.40 1.50 2.00 3.00811 4.00 5.00 6.00 6.70 6.75816 6.80 6.810 6.820 11.10 11.150821 11.203 11.30 11.50 12.00 12.50826 13.00 13.50 13.983 14.I0 14.20831 14.30 14.40 14.50 14.60 14.70836 14.80 16.9360
X COORDINATES OF THE BODY PANEL CROSS SECTIONS (R INTERPOLATED)851 6.1839
862 14.0775
R VALUES OF THE BODY GEOMETRY SECTIONS.901 O. 0.250501 0.265452 0.267113 0.283416906 0.299132 0.314302 0.328960 0.395480 0.501051911 0.577263 0.628500 0.656942 0.663870 0.663961916 0,663999 0.664 0.664 0.664 0.664921 0.664 0.663791 0.662036 0.649816 0.626208926 0.590771 0.542754 0.483287 0.466763 0.451935931 0.436431 0.420224 0.403653 0.387083 0.370513936 0.353943 0.0
X COORDINATES OF THE AXIS SINGULARITIES.I001 0.0 0.75 1.25
1031 0.0 15.75 16.25 16.9360X COORDINATES OF THE AXIAL SINGULARITY CONTROL POINTS
1101 1.00 1.50 2.001131 16.00 16.50
10.
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89 0.05 -1.0ROOT ( < 0. INDICATES LEADING AND TRAILING EDGE INPUT)
103-9.00< 0. INDICATE PANEL SPACING IS EVEN.
107-1.0CORDINATES OF PANEL ENDS
161 0.6640 0.9323172 3.2215CORDINATES OF LEADING EDGE
241 6.8072
252 13.8333CORDI NATES OF TRAILING EDGE
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1 1.0= -I.0 NO MORE CASES FOLLOW.
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OPTI HI ZATI ON PROGRAM OPT
DISCUSSION
The computer program OPT obtains the optimum camber, twist, and/or flap
deflections for a given configuration by obtaining the minimum zero suction
drag subject to various constraints. On the lifting surface panels, the
component of force in the drag direction is equal to the local angle of
incidence times the force in the direction normal to the panel. The zerosuction drag is thus defined as the sum:
Cd * SREF =N
sum Cp(i) * Alpha(i) * Pa(i)i=l
The index i runs over all N finite elements (panels) into which theaerodynamic surfaces are divided, and
Cp(i)Alpha(i)Pa(i)
Is the delta Cp across element iIs the local angle of attack of element iIs the area of element i
All pressures and velocities in program OPT are computed usingsubroutines from program WBODYwhich, for first order analysis, uses all ofthe assumptions of linearized, thin wing theory. Optimizations may beperformed at all Mach numbers on configurations having up to ten surfaces, andone body. The program does not compute configuration geometry or aerodynamicinfluence matricies. This information is read from data files which are
created by other programs such as WBODY or APAS.
The final optimized result may be constrained to maintain a total liftcoefficient and center of pressure, and various combinations of liftcoefficients on any specified set of surfaces, spanvise section lift
coefficients, spanvise variation of sectional center of pressure (x/c-C.P.),spaneise twist variation, twist and camber over specified span stations, andflap deflection angle.
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For a first order optimization, with or vithout a paneled body, twodifferent methods are used. The first uses constraint functions for chordwise
pressure distributions at any or all span stations. Span stations having nochordwise pressure constraint functions may be thought of as actually having anumber of constraint functions equal to the number of chordvise panels. Eachfunction is then a delta function which is equal to 1.0 on the panel equal tothe constraint function number, and equal to 0.0 elsewhere.
The second method uses chordeise polynomial curves, and flap deflectionslopes over selected panels as camber constraint functions. The solution ineach case consists of the coefficients of these constraint functions at each
span station. If a paneled body is present, since its' geometrical shape isfixed, the only degree of freedom allowed for it is the angle of attack, whichmay be constrained to a fixed value if desired.
Second order optimizations may be performed, using camber constraintfunctions on planar configurations without a body. The influence of thicknessmust be considered, otherwise the result will be the same as a first orderresult.
Cases using chordvise pressure constraint functions, or using noconstraint functions, will be referred to as a Cp optimization. This methodof solution first yields panel strengths (delta-Cp's on vortex panels andsource strengths on body source panels) which result in minimum drag, subjectto any constraints, and then the camber and twist is determined from the panelstrengths and the aerodynamic influence matrix.
When using camber constraint'functions, the optimization will be called atwist optimization, which is a shortened form of saying twist and camberoptimization. This name originates because the solution yields the twist andcamber shapes which result in minimum drag, subject to any constraints. Thepanel strengths (delta-Cp's on vortex panels and source strengths on bodysource panels) may be obtained by solving the set of simultaneous equationsusing the aerodynamic influence matrix. The simplest case using this methodwould be a solution which considered only the lowest order camber functions,consisting of uncambered constraint functions (first order polynomials) ateach span station. This would mean the angle of attack or twist at each spanstation was the only degree of freedom, and the solution would consist of theoptimum distribution of twist with vhatever initial camber was input.
The program uses geometry data, aerodynamic influence matricies, and thequasi-inverse matrix (if present), from either computer program WBODY, APAS orUDP. These data are read from datasets which are allocated to the correct
unit numbers.
The program may also be used for analysis only using the geometry andaerodynamic influence matricies read from the respective datasets.
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INPUT DATA
Input data, which directs the program operation, is read using subroutineDECRD1, described on page 19, and is stored in the array called "DATA". All
locations are initially set equal to 0. Jet flap and vortex lift capabilityare not available in the program at this time and any input which refers tothese capabilities could cause unpredictable results.
There are three different types of program operation, a Cp optimization,a twist optimization, and an analysis. The following is a brief descriptionof the program input necessary for these three different modes. A moredetailed description of all input locations follows.
Cp OPTIMIZATION
This option may be used to find optimum twist and camber subject tovarious constraints. Span stations may be designated where a specified camberis to be maintained, or where a specified chordwise Cp distribution is to bemaintained. The unknowns consist of panel Cp's, at span stations where thereare no constraint functions, and the coefficients of chordwise Cp functions atspan stations where there are constraint functions. The camber is obtained bymultiplying the optimized Cp distribution by the aerodynamic influence matrix.
The following data must be input for a Cp optimization.
Input variables location
Total CL
XCG - the desired center of pressure
432
The followin 9 input is optional
Input variables location default
The number of chordwise Cp constraint functions 5The type of aero matrix and inverse 7
Second order anlysis of optimized result 8Roll, yaw, and side force constraints (asymmetric) llReference span station for angle of attack 21Input camber 29Mach number for suction calculation 35
x/c for n-velocity in drag analysis & optimization 36CBAR 40C_VG 41SREF 42
20.
nonenone
1
none
geometry0.515,0.875
geometrygeometrygeometry
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SPAH 43
Additional angles of attack 52Additional angles of attack (unit solutions) 62Individual surface CL's 71
Fixed camber, or fixed Cp span stations 81 +Desired CL*c/Cavg at span stations 151 +Relative delta CL at span stations 201 +x/c of center of pressure (C.P.) at span stations 251 +Relative delta x/c-C.P, at span stations 301 +Relative delta CL at span stations 351 +Span stations with no constraint functions 401 +x/c's to input camber (method J 3, see loc 29) 551 +z/c's to input camber (method J 3, see loc 29) 601 +Twist constraint reference stations 951 +
Additional input camber (method J 2, see loc 29),
m or fixed chordwise Cp distributionsFlap deflections (input camber)
The default "geometry" means the values from the
I dataset which contain the geometry are used.
2840 +3340 +
geometrynone
nonenone
nonenone
none
nonenonenone
see 5
nonenone
none
nonenone
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The following input data controls the output which is printed
Type of Printout location default
Constraint equation printoutThickness z/c, dz/dx, and initial camberGeometryAdditional printoutPlot dataset created
14 none15 none16 some19 some20 none
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TWIST 0PTIMI ZATION
This option may be used to find optimum tvist camber and flapdeflections. The constraint functions consist of camber geometry shapes,vhich result in a series of unit solutions, and the optimization consists of
finding the coefficients of these unit solutions. The Cp's are obtained bysolving a set of simultaneous equations using the aerodynamic influencematrix.
The final wing camber will be equal to the input ring camber (see datalocation 29) plus the appropriate sum of the camber constraint functions (ifany) plus any flap defections.
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The following data must be input for a twist optimization.
Input variables location
Total CL
Twist optimization flag > 0.XCG - the desired center of pressure
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32
The following input is optional
Input variables location default
The number of camber constraint functions 5 1
The type of aero matrix and inverse 7 0.Second order optimization or analysis 8 noneRoll, yaw, and side force constraints (asymmetric) 11 noneTrimmed or not trimmed 12 not
Reference span station for angle of attack 21 1Type of interpolation for flaps 23 noneInput camber 29 none
Mach number for suction calculation 35 geometryx/c for n-velocity in drag calculation and OPT 36 0.515,0.875
CBAR 40 geometryCAVG 41 geometrySREF 42 geometrySPAN 43 geometryAdditional angles of attack 52 none
Additional angles of attack (unit solutions) 62 noneIndividual surface CL's 71 none
Flap panel locations 81 + noneConstraints on camber functions 151 + noneConstraints on twist 351 + none
Zero input camber at span stations 401 + none
x/c's to input camber {method I 3, see loc 29) 551 + none
z/c's to input camber {method I 3, see loc 29) 601 + noneTvist constraint reference stations 951 + none
kdditional input camber {method J 2, see loc 29) 2840 + 1.0
Flap panel ratios and deflections 3340 + none
The default "geometry" means the values from the
dataset which contain the geometry are used.
The print control input is the same as for a Cp optimization.
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ANALYSIS
The program may be used to analyze a given configuration at variousangles of attack with or without camber. The program uses the specified inputgeometry and aerodynamic influence matrix. The Cp's at specified spanstations may be set equal to zero in order to find the influence of various
span stations or surfaces on other span stations or surfaces.
The following data input is required for an analysis.
Input variable location
Analysis flag = 1.0 12
The following input is optional
Input variables location default
The type of aero matrix and inverse 7Second order analysis 8Roll, yaw, and side force constraints (aslmmetric) ll
.
nonenone
Reference span station for angle of attack 21 1?ype of interpolation for flaps 23 noneInput camber 29 noneMach number for suction calculation 35 geometryx/c for n-velocity in drag calculation 36 0.515CBAR 40 geometryCAVG 41 geometrySREF 42 geometrySpan 43 geometryAngle of attack 51 add loadUnit solutions 61 41.02Surface CL's 71 none
Span stations to zero Cp or camber 81 + noneAdditional input twist 351 + nonex/c's to input camber (method # 3, see 29) 551 + nonez/c's to input camber (method | 3, see 29) 601 + noneTwist constraint reference stations 951 + none
Additional input camber (method # 2, see 29) 2840 + 1.0Flap panel ratios and deflections 3340 + none
The default WgeometryW means the values from thedataset which contain the geometry are used.
The print control input is the same as for a Cp optimization.
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DETAILED DESCRIPTION FOR DATA INPUT
Location Data
If this value is nonzero the case is terminated
4 The total CL of all surfaces.
5 l.J NFX = I = The number of chordwise constraint functionsNFY = J = The number of spanwise constraint functions
Two cases:
1. DATA(17) : 0. Called a Cp optimization.Cp optimized - camber results from Cp
I=0
l<0NFX = 2 is used as a default value.No constraint functions are used.
Constraint functions should always be used forsubsonic Mach numbers.
Individual span stations may be specified where noconstraint functions are desired using datalocations beginning at 401.
Fixed span stations (see location 81) always haveno constraint functions.
NFY is ignored (NFT = 0 is used regardless of input)
2. DATA(17) > 0. Called a twist optimization.
Camber optimized - Cp results from camber
If0I>1
Twist only is optimized. (same as I = l)Twist + (l-l) camber constraint are used.
If J = NFY > 0 (works only with a single surface)
The twist (or camber constraint coefficient) values,U(k,eta), are constrained spanwise on each surfaceIn the form:
NFY
U(k,eta) = stun a(k,i+l) * eta**ii=0
k = 1,NFX
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IMATRX. JMATRX
IMATRX < 0
IMATRX < - I
IMATRX < - 2
If the aeromatrix is from WBODY.If the last aero matrix is an inverse matrix.If 1st aero matrix is a thickness matrix.
JMATRX> 0
JMATP,X"
This is used only with configurations havingbodys which are paneled with source panels.
Bodlmx Body Cp Un (for body configurations only)
0 Yes No No1 Yes No Yes2 Yes Yes Yes
3-4 Half No No5 No No No6 No Yes No
7-9 No No No
(does constraint functions)
< 0. Second order analysis.-1. Second order optimization is performed.
< -1. Second order optimization, including thickness drag.A second order optimization can be used with DATA(17) > 0. only
The value of q (dynamic pressure). This input is used tosignal that a flexible configuration is being considered.This can be used only with a twist optimization(DATA(17) > 0.).
The number of g's at the design CL (DATA(4)). This value is
used to multiply the weights from locations 1801-2301 inorder to find the deflections due to the inertial loads on
a flexible configuration.
If 0. The symmetry data from the geometry dataset is ignored.
= 1. Constraint equation included for roll.= 2. Constraint equations included for roll and yaw.= 3. Constraint equations included for roll yaw and side force.> 3. Uses slmmetry data with no additional constraints.
These constraints are used only for aslmBetric configurationsXCG, YCG and ZCG obtained from DATA(32-34)
If a negative number is input, a solution is obtained forflap settings to satisfy a roll constraint for a flexibleconfiguration. When applied to symmetric configurationsan anti-sI_netric aero matrix should be used. If q = 0. isdesired (DATA(9)), set q< 0.
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If<-2.If=-2.If---1.If" O.If = 1.If> 1.
Prints geometry only, no optimization or analysis.No add load and no drag polar.If DATA(17) > 0. , calculates trimmed polar.An add load and an optimization is computed.Computation using input cambers (analysis only).Computation using Cp's from plot dataset (WBODY).
MlX.NUX MIX ffi the number of constraint equations printedNUX ffi the nmnber of Q(I,J) (drag matrix) rows printed.
If greater than the number of equations, all are printed.If a negative number is input the equations will be printed
Only if the matrix is singular.
= 1. Print source dz/dx and z/c matricies
= 2. Print both source and camber arrays= 3. Print camber slope matrix
The geometry printoutIf > 0. or < -1. A wide range of geometric parameters will
be printed.
If > 0. A twist optimization will be performed,unless DATA(12) > 0.
It may be recalled that a twist optimization is only aterm used to des_nate the type of optimization procedureinvolved. The twist optimization procedure may be used tofind optimum twist camber and flap deflections. The
following input should be considered when performing atwist optimization.
The final wing camber will be equal to the input wing camber(see discussion for data location 29) plus the appropriatesum of the camber constraint functions (if any) plus anyflap defections.
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Printout of velocities due to thickness (source panels).
>1.2.
>3.>4.>5.
printout of w velocities (normal to panel).printout of u velocities (x direction).printout of v velocities (bi-normal to panel).printout of z/c for thickness.printout of dw/dx (2nd order only).
IFt_ED Various orders of intermediate printout (-2. to 4.)
-2.-1.
2.
.
4.
Least printout (no drag n-velocities)No upper and lower surface cpsMore printout
Prints flexible unit solutions and
(v-w), (w-b), (b-v), (b-b)/d_ove +Above + 2nd order boundary condition solutions
DPL_ O.1.2.
>2.
No data is placed in a plot dataset.Data is placed in an i_PAS plot dataset (geometry).Data is placed in a UDP plot dataset.Data is placed in a UDP plot dataset (extended).
The reference span station which will be used to calculatethe angle of attack. This angle of attack is only usedfor reference purposes in order to calculate twist. IfA0(K) and A0(J) are the local angles of attack withrespect to the ft,,stream, COSZ(J) and COSZ(K) are the localdihedral angles, and K is the reference span station, then:
Alpha = XO(X) / COSZ(X)
and the value of twist for span station J wili be:
TWIST(J) = XO(J) - klpha * COSZ(J)
X<O
K=O
K>O
All twist values will be referenced to the freestream,
i.e. Alpha = 0.K = 1 is used. If a body is present then Alpha is
determined by the angle of attack of the body.This station is used as the reference span station.
determined by the angle of attack of the body.
<0. All normal velocities (slopes), used to compute the zero
suction drag, are interpolated chordwise from theboundary condition control points (usually x/c = 0.875of each panel) to x/c = RD (see DATA(36)). A thirdorder curve f_t is used for this purpose.
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29
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If>0
Otherwise (default), the normal velocities are interpolatedassuming possible discontinuities where flap panelsoccur (see locations 81-150), and the chordwiseinterpolation of normal velocities is performed whilethe flap deflections are removed. The interpolation isperformed using information obtained in data location36.
Surface to be extended to centerline (paneled bodies only)
The surface will only be extended when the aeroinfluence matrix is calculated.
S = surface number (in ascending order).
K=OK=IK=2
Extend inboard panels with zero sweep.Extend inboard panels with same sweep.Extend inboard panels with negative sweep.
i.e. if DATA(26) = 2.0 The calculation of the aeroinfluence matrix for surface I 2 will be
performed by extending the inboard panels toy - 0. with zero sweep,
N.MF (N,M,F integers) Initial camber input specification.
N < 0 No initial camber is assumed.
There are three ways of inputing any initial camber andthe resulting camber slopes from each method will beadded together.
. Reading span station twists and camber normalvelocities at panel control points from thedataset containing the geometry.
. Inputing normal velocities at panel control pointsusing locations beginning at 2840 or 3340. Forcases where it is appropriate, values of twistmay be input using data locations beginning at351.
3. Inputing x/c and z/c using locations beginning at3951 and 4001.
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method # 1
If M > O, the casber will be read from the geometrydataset in the following manner.
M=I The camber values only (not the twist or angleof attack) are to be read from the geometrydataset. Any twist in the initial camber isremoved.
Mffi2 The camber values only (not the twist or angleof attack) are to be read from the geometrydataset.
M=3 The twist and camber values (no angle of attack)are to be read from the geometry dataset.
Mffi4 The twist, camber and angle of attack are tobe read from the geometry dataset.
If F k 0, the camber will be read from the geometrydatas_t in the following manner.
F_2 The flap values will be read from the geometrydataset and subtracted from the initial camber.
F>4 After subtracting from the initial camber, any
flap deflections from the geometry datasetare set equal to zero.
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method | 2
Unless N < 0, twist and camber values may be obtainedfrom normal velocities (normal facing upward from thepanel) input in the order of the control points
beginning in locations 2840 and 3340. The values inputin these locations may include both twist and camber.If no values are input, there will be no be no camberinput by this method, since all data locations areinitially set equal to zero for the first case. Thevalues from the 2840's are assumed to represent a
smooth camber distribution and will be interpolated tocompute drag. The values from the 3340's are assumedto be due to flap deflections and will not beinterpolated when drag is computed (see DATA(24)).
method # 3
If N > 0 camber may be input in the following manner.
N = ND > 2, the number of x/c values where the z/c valuesfor camber are specified (locations 3951,4001). Thesex/c stations are the same for all span stations andmust include x/c = 0. and x/c = 1. The camber, andz/c's apply to fixed span stations only (see DATA(81)),unless DATA(17) > 0.
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3839
XO0
XCG
YCG
ZCG
RC
RS
The • value of the pivot point for angle of attack.
Used for second order theory only.
The desired center of pressure (to override value fromthe geometry dataset. If a value < -99999. is input theXCG constraint is not used.
For non-slnnnmtric rolling moment only (see DATA(If))
For non-slmunetric rolling moment only (see DATA(ll))
The freestream Much number (used for computing leadingedge suction only).
If 0. the value from the geometry dataset will be used.If < 0. the Much number will be set equal to 0.
The x/c on each panel where the normal velocities (localangles of incidence) are interpolated for optimizing dragand for computing drag.
If RC< 0.IfRC= 0.If RC> 0.
oPr @ x/c =- RCOPT @ x/c = 0.875OPT @ x/c = 0.875
analysis @ x/c = 0.515analysis @ x/c = 0.515analysis @ x/c = RC
Recall that the drag increment from each panel is equal
to the delta Cp times the area times the local angle ofincidence. The local angle of incidence may vary znthe chordwise direction, and the boundary condition issatisfied on each panel only at x/c = 0.875. Thereforethe interpolation is necessary to obtain an angle ofincidence for each panel which is a closerapproximation to the average value for the panel.
The x/c fraction of each panel which is used tocurvefit (x,¢p(x)) in order to obtain an approximatevalue for the leading edge suction (default = 0.250).
Since RS < 1.0, if the value is input as M.RS, withM k 1, M will be the number of functions used for thecurvefit. If M = 0, the default value is 3 functions(i,e. M = 3). If < 0. a detailed printout results.
Drag polar (41 points given)
Delta CL for drag polar. Default = 0.05Starting CL for drag polar. Default = 0.0
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5O
51
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61
CBAR The reference chord length for computing QnIf 0. CB_ = CAVG is used.
CAVG The reference chord length.
If 0. value from the geometry dataset is used.
SREF The reference area.
If 0. value from the geometry dataset is used.
Span Span, any consistent set of units may be used. Thisvalue is used for reference purposes only.If 0. value from the geometry dataset is used.
Initial guess for angle of attack (for configurationshaving bodys paneled with source panels)
kLPH&D(1) The configuration angle of attack with respect to the
freestream (degrees). This is in addition to anycamber distribution. For an optimization theresulting angle of attack is used for ALPItkD(1). Ifa value is input for &LPH/d)(1), and a body ispresent, it will be assumed to be a constraint onthe body angle of attack.
ALPHkD(2)_PHAD(8)
cA_mn_(1)
The 2nd angle of attach where an analysis is desired.(maximum number of angles of attack = 8)
If ALPItkD(K) < -90. kLPH_D(K-1) is used.
If ALPH_D(K) > 90. an add load solution is given.
i.e. (Alpha = 1.0) - (Alpha = 0.0)
Input in form OB-OV.CkM.THK, and used with _LPHAD(1)
This input is used to define which set of unitsolutions to combine into a final solution. Since
the solution is linear, the unit solutions maybe combined in any combination.
OB = the order of the Cp on the body (one digit).OV = the order of the Cp on the lifting surfaces).CAM > 1 camber included.THK > 1 thickness included.C_M = 0 no camber included.
= 0 no thickness included.
0B, 0_, CAM, THK are each one digit.
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If<0.
OB = 4, is used for the body Cp,OV, is determined by DkTA(8), and
camber and thickness are included.
OV = 2, if DATA(8) < 0.
OB
Same as above except the 2nd order axialsolution (if performed) is used tocalculate u,v,w.
Cp
12345
The body pressure formula isdetermined by the value of OB.
Cp=-2tuCp = - 2 * u- beta2 * u'u- v'v- w*wCp = - 2 t u - utu - vtv - wtwCp = Isentropic pressure formula (default)Cp = Isentropic for Alpha = O. + isentropic add load.
u,v,w Use 1st order axial contributions if camthk > 0u,v,w Use 2nd order axial contributions if camthk < 0
E.9. c_WrHK = - 31.02
- Indicates, 2nd order u,v,w from axial solution (if performed).3 indicates, pressure formula |3 on body.I indicates, Ist order Cp on lifting surfaces.0 indicates, camber is not included.2 indicates, thickness is included.
62 CAMT_(2) Used with /_PHAD(2)68 CAMTHK(8) Used with ALPH/LD(8)
71 The desired values of CL for each surface. (maximum of 10)If 0. the CL will not be constrained for that surface.
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81 The input depends on whether a Cp optimization, a twistoptimization, or an analysis is to be performed.
* * * For a Cp optimization (DATA(17) = 0. ) • , ,
The input consists of the list of span station numbers wherea fixed camber is desired, followed by a list of spanstations (input as negative numbers) where a fixed Cpdistribution is desired.
The span stations where a fixed camber is desired arereferred to as fixed span stations, and the resultingcamber slopes will be equal to the initial input camber.The list of span station numbers must be in ascendingorder.
These span stations have the following properties:
. If a span station number in the list is input as J.K,K > 0, rather than just J., then the J'th span stationis assumed to be a fixed span station with an unknownvalue of twist (see 351 for a definition of twist).
All subsequent fixed span stations will have this valueof twist (plus any twist difference specified by theinital input camber, which may contain twist, or thevalues obtained from locations beginning at 351), untilanother J.K is found in the list of fixed spanstations. (e.g. this option could be used for a fixedsurface with an unknown deflection).
. There are no Cp constraint functions used. Each panel Cpis regarded as an unknown (this can be thought of as adelta function type constraint function).
3. The initial twist and camber is used (see the discussionfor data location 29).
. The final slopes are obtained by adding the optimizedangle of attack to any initial input twist andcamber.
5. No x/c-C.P, constraint is allowed.
6. No CL*c/CAVG constraint is allowed.
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The span stations where a fixed Cp distribution is desiredare input as negative numbers following the list of spanstations where a fixed camber is desired. The list of spanstation numbers must be in ascending order. The Cpdistribution for this span station is input in place ofnormal velocities in locations beginning at 2840.
i.e. If the span station has panel numbers beginning withijl and ending with ij2, the Cp's are input in locations(2840 + ijl - 1) thru (2840 + ij2 - 1).
These span stations have the following properties:
. There is one Cp constraint functions used at each ofthese stations. This constraint function has the
desired Cp distribution.
. If a span station number in the list is input as - J.K,K > 0, rather than just - J., then the J'th span
station will have the desired Cp distribution
multiplied by a constant to be determined during the
optimization.
3. No x/c-C.P, constraint is allowed.
4. No CL*c/CAVG constraint is allowed.
. Twist constraints may be used in the same manner aswith span stations which do not have a fixed camber(see the above discussion of fixed span stations andthe discussion of twist constraints at data location
351).
* * * For a twist optimization (DATA(17) > 0. ) * * *
These locations are used to define the panels where flapsare located. It may be recalled that a twist
i optimization is only a term used to designate the typeof optimization procedure involved. The twistoptimization procedure may be used to find optimumtwist camber and flap deflections. Further properties
I of these flaps (deflection ratios) may be input inlocations beginning with 3340.
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le An input value of J.MMNNKK means that for span station
J flap number KK runs from panels MM to panels nn.
If KK is input as 00 it will be set equal to 01.A maximmaof 20 flaps are allowed (KK=20).
* * * For an analysis (DATA(12) = Io ) * * *
For an analysis these locations are used to specify thespan stations where the Cp's are to be set equal tozero, or the normal velocities on all panels of thespan station are to be set equal to zero.
This feature may be used to find the normal velocities
induced on any set of span stations (or surfaces) bythe remaining set of span stations (or surfaces). The
normal velocities induced at the panel control pointsby all span stations where the panel Cp's were not setequal to zero may be found under the heading "DRAGN-VELOCITIES" in the printed output. Angle of attackand the presence or absence of camber may be controlledusing locations 51-58 and 61-68. Data location 36
should be set equal to 0.575 to supress theinterpolation of the normal velocities to other than
the panel control point (x/c = 0.875). The averageincidence of induced velocities, for the first angleof attack only, may be found under the twist column
in the printed output, or in the plot output datasetas twist.
The span station numbers should be input in ascendingorder.
. A span station number input as a positive number willcause the panel Cp's to be set equal to 0.0 on thatspan station. The panel Cp's on all remaining spanstations (except those as described below) will remainat the value achieved after an analysis at thespecified angle of attack, and with or without thepresence of camber.
. A span station number input as a negative number willresult in the normal velocities on each panel of thatspan station to be equal to 0.0. The Cp's on allspan stations input as negative numbers are adjustedto achieve this result by canceling the normalvelocities induced by all panels where the Cp's werenot set equal to 0.0.
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201
251
301
'kt
_t
For a Cp optimization (DATX(1?) = 0.) t_
The desired values of CL*C/CAVG at each span station.If 0. is input, the value is unconstrained.
For a tvist optimization (DATA(17) > 0.) **
Used in conjunction vith DATA(5), these locations areused to fix the coefficients of the camber constraint
functions to desired values (input < 1.), or to constrainthe value to be equal to the value at another spanstation (input the other span station number • 1.). Ifleft 0., the coefficient value is unconstrained and anoptimized value will be chosen. Only the camberconstraints are input at these locations. For twistconstraints see location 351.
If a camber constraint coefficient of a given order isconstrained, all higher order coefficients at the samestation are also constrained, even if zero is input.
Input is in order of the functions at each span station.
151152
150+(I_X-1)
1st camber function for station # 12nd. camber function for station | 1
1st camber function for station # 2
2nd camber function for station # 2
Locations 201 - 350 are used for Cp optimizations only.
The relative value of delta CL at each span station
If no values are input, the constraints will besatisfied exactly.
The desired values of x/c-C.P. At each span station
If 0. is input, the value is unconstrained
The relative value of delta x/c-C.P, at span stationsAt least one of these values must be nonzero if any x/c-C.P.values are asked for. If a uniform change in x/c-C.P, isdesired the value in these locations should be equal to the
expected value of CL*c/CAVG.
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351 These locations are used for inputing initial twist values or
twist constraints. The input in these locations depends on
whether a Cp optimization, a twist optimization, or an
analysis is being performed. The twist of a span station isdefined below.
* * * For a Cp optimization (DATA(17) = 0. ) * * *
If span station J is a fixed span station, the valuesinput in DhTA(350+J) are treated as part of the initialtwist input for span station J. Therefore if spanstation JP is the immediately preceeding fixed spanstation (to span station J), which had an unknown value
of twist (J.K in DATA(SI-150)), the resultingoptimization will have:
TWIST(J) = TWIST(JP) + (TWIST0(J) - TWIST0(JP) )
where TWIST0(J) and TglST0(JP) are the initial twist
input values for span stations J, and, JPrespectively (see data(29) for the input of initialtwist and camber.
If J = 1 the angle of attack of the first span stationwill be equal to DATA(351) plus any initial angle ofattack (from locations beginning at 2840 and 3340 andthe geometry input dataset (fixed span stations only).
If span station J is not a fixed span station, any valuesinput in DATA(350+J) are will be used to constrain the
twist diffe¢ence between span station, J, and areference span station, JR. If DATA(350+J) = 0. thevalue of TWIST(J) will be unconstrained. Therefore the
resulting optimization will have:
TWIST(J) = DATA(350+J) + TWIST(JR)
JR = DATA(950+J)
if JR = 0
if JR > 0
if JR < 0
JR = J-I is usedJR = JR is used
the constraint is made with respectto the free stream, i.e.
TWIST(J) = DATA(350+J)
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For a twist optimization (DATA(I7) > 0. ) * * *
If DATA(350+J) = 0., then the twist of span station J willnot be constrained.
Any values input in DATA(350+J) will be used to constrain
the twist difference between span station, J, and areference span station, JR. If the initial twist of
station J was set equal to zero (see data(401)), thetwist constraint is mde on the final twist difference
between the stations. If the initial twist was not set
equal to zero, the constraint is made on the differencein the twist increments which are added to the initial
twists at the two span stations.
final twist difference constraint
TWIST(J) = DATA(350+J) + TWIST(JR) + TWIST0(JR)
constraint on increment
TWIST(J) = DATA(350+J) + TWIST(JR)
Where:
TWIST0(L) is the initial input twist of any station L
TWIST(L) is the increment of twist give to station L
TWIST(L) + TWIST0(L) is the final twist of station L
JR = DATA(950+J)
if JR= 0if JR> 0if JR<O
JR = J-1 is usedJR = JR is used
the constraint is made with respectto the free stream, i.e.
TWIST(J) = DATA(350+J)
If J = 1 the angle of attack of the first span stationwill be equal to DATA(351) plus any initial angle ofattack (from locations beginning at 2840 and 3340 and
the geometry input dataset.
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* * * For an analysis (DATA(12) = 1. ) * * *
TWIST(J) = DATA(350+J) + Twist0(J)
i.e. the initial twist of span station J isIncreased by a fixed increment (DATA(350+J)).
The twist of a given span station is defined as the angle ofattack of the span station (with any camber removed) withrespect to the freestream, when the angle of attack of theconfiguration is zero. This means the twist of span station Jis equal to the local angle of attack with respect to thefreestream, A0(J), reduced by the angle of attack, Alpha,times the local dihedral angle, COSZ(J).
i.e. TWIST(J) = A0(J) - Alpha * COSZ(J)
Alpha Is either the angle of attack of the referencespan station or zero (see DATA(21)).
The angle of attack, Alpha, may be obtained from the localangle of attack, A0(K), of the reference span station K,which is measured with respect to the freestream.
Alpha = - A0(K) / COSZ(K)
e.g. Suppose reference span station K and span station Jhave local angles of attack A0(K), and A0(J), with
both A0(J) and A0(K) measured with respect to thefreestream. Let COSZ(K) and COSZ(J) be the local
dihedral angles. Then if a(K) and A(J) are the
respective angles of attack after a pitch angle Alpha:
A(K) = A0(K) + Alpha * COSZ(K)
A(J) = A0(J) + Alpha * COSZ(J)
The twist of span station J is defined as the local angleof attack, A(J), when the reference angle of attack,
A(K) = 0. Therefore using the above:
Alpha = - A0(K) / COSZ(K) , and
TWIST(J) = A0(J) - A0(K) * COSZ(J) / COS(Z)
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401
551
601
** Fcr a Cp optimization **
/t_t
The span stations where no constraint functions aredesired. See also location 5.
For a twist optimization **
Input as J.KL, where J is the span station where the camberor twist from the geometry dataset are to be set equal tozero (see location 29 for a discussion of input camber).
If K = 0 the camber values are set equal to zero.If L = 0 the twist values are set equal to zero.
e.g. an input value of 12.02 means that span station 12will have the camber, but not the twist, from the inputdataset set equal to zero.
The J values (span stations) must input in ascendingorder.
The x/c values where the camber is specified (see 601).
The z/c values for camber.
The values for x/c = 0. and 1. must be specified.
The z/c values for each x/c at every span station areinput in consecutive locations starting with location451. The values are input first in the order of thex/c values for a given span station, and next in theorder of the span stations. Any location left blankwill be assumed to have a z/c value of 0.
Camber values should be input only for fixed spanstations (location 81), or when a twist optimizationis preformed (location 17). Twist values may beinput using locations 351 to 400.
If an analysis only is being performed, data location 17
should be made • 0. In this case all span stations mustbe input (although 0. is permissible). This is to avoidconfusion of the fixed span stations with span stationswhere the Cp is to be set equal to zero (see thediscussion for an analysis under location 81).
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951
100110021003
100410051006
1007100810091010
Twist constraint stations.
L = DATA(950+J}
Twist constraint at span station J implies a
nonzero of DATA(350+J)o
L < 0 Any twist constraint at span station j will bemade with respect to the free stream.
L = 0 Any twist constraint at span station j will bemade with respect to span station (j-l).If j=l the constraint will be with respect tothe freestream unless a body is present.
L > 0 Any twist constraint at span station j will bemade with respect to span station L.
L > nst Any twist constraint at span station j will bemade with respect to the body.
Hinge Homents locations 1001 - 1100
The values in the following locations are for calculating orconstraining up to twenty-five hinge moment coefficients. Thehinge moments are calculated by taking the sum of the momentsinduced about a given line by a set of prescribed panels.Hinge moments may be constrained only when performing a twistoptimization (DATA(17) > 0.).
The two points which determine each line about which the moments
are taken, are input in locations beginning at 1001. Thepanels determining each integration area are input inlocations beginning at 1051.
The x value of the 1st point for hinge moment line number 1.The y value of the 1st point for hinge moment line number 1.The z value of the 1st point for hinge moment line number 1.
The z value of the 2nd point for hinge moment line number 1.The y value of the 2nd point for hinge moment line number 1.The z value of the 2nd point for hinge moment line number 1.
CBAR for hinge moment number 1 (if 0. uses CAVG).The constrained value for this hinge moment (no constraint if 0)if < 0. hinge moment array number 1 will not be printed.Reference area for hinge moment number 1 (if 0. uses SREF).
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101110121013
101410151016
1017101810191010
1051
1200
1201
1801
The x value of the Ist point for hinge moment line number 2.
The y value of the 1st point for hinge moment line number 2.
The z value of the 1st point for hinge moment line number 2.
The x value of the 2ndpoint for hinge moment line number 2.The y value of the 2ndpoint for hinge moment line number 2.The z value of the 2ndpoint for hinge moment line number 2.
CBhR for hinge moment number 2 (if 0. uses CAVG).
The constrained value for this hinge moment (no constraint if 0)
if < 0. hinge moment arra_ number 2 will not be printed.
Reference area for hinge moment number I (if 0. uses SREF).
Beginning in this location the panel numbers for determiningeach hinge moment are listed. An input value of 0. indicatesthe last panel number for a given hinge moment calculation hasbeen input. A sequence of panel numbers may be input byinputing a negative panel number after a positive number.
i.e. 1051 3 5 10 20- 55 0 3 - 40
will have panel numbers 3,5,10, and 20 thru 55 for #1and panel numbers 3 thru 40 for t2
due to camber, input in same order as the control points.These values will added to camber values input by othermeans. These values are assumed to represent a smoothcamber distribution and will be interpolated to compute
drag.
If > 0. The effect of the normal velocities due to thickness
will be included in the optimization.
The normal velocities at the panel control points due tothickness.
The total weight distributed over each panel. These valuesare used in conjunction with location 10 to obtain thedeflections induced by the inertial loads on a flexibleconfiguration.
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2840
3340
The normal (outward) velocities at the control pointsdue to camber, input in same order as the control points.These values will added to camber values input b_ othermeans. These values are assumed to represent a smooth
camber distribution and will be interpolated to computedrag.
These locations are also used to input a desired Cp distributionfor a given span station (see the discussion for data(81)).
i.e. If the span station with a desired Cp distribution haspanel numbers beginning with ijl and ending with ij2, the Cp'sare input in locations (2840 + ijl - 1) thru (2840 + ij2 - 1).
The same as for 2840 except the values are assumed to bedue to flap deflections and will not be interpolated whendrag is computed, nor will the the deflections contribute
to camber when computing twist.
If flap panel ratios are desired for a twist and flapoptimization, (DATA(17) > 0.), the flap panel ratios are
input in these locations. I.e. if panel IJ is a panel offlap # j, and it is desired that this panel will deflect0.60 units for each unit deflection of flap # j, then set:
Data(3340+IJ-1) = 100.60
The value of 100. is added to differentiate between initial
deflections and flap panel ratios. It is assumed thatpanel IJ is defined as a flap panel through locations 81-160.
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LIST OF VARIABLES
OPTIMI ZATION VARIABLES
NFX
NFJ
The number of constraint functions at each station
The number of constraint functions at each station (jet)
ISFX
ISFJ
The surface number where the jet attaches(if ISFX = NSF the jet is removed)
The surface number of the jet(if ISFJ .gt. NSF) There is no jet
ITRIM,]TRIMKTRIM
The character array number for trim variation variableThe constraint equation number for the trim variation.The variable which determines the desired variable for
trim variation.
NCLSMCLYNPBNSTBMTWlSTMUTWSTMXCP
The number of surface CL constraints
The number of spanviseCL constraintsThe number of panels on fixed span stationsThe number of span stations without constraint functionsThe number of twist constraints
The number of unknown twists on fixed span stationsThe number of x/c-C.P, constraints
CXX(IJ)WF(IJ)
M1M2HUMLT
The normal velocity at the control point due to camberThe deflections at the control points due to flaps.
The number of equations to be solved exactlyThe number of equations to be solved least squareThe total number of unknownsThe total number of B(ML)
GENERAL VARIABLES
cxom}(a)IJS(IS)iao(a)ISO(ISF)JS(IS)MS(IS)HBHCHORD(J)NSNSFNSP_(ISF)NSLHST
The chord value at the centroid of span station JThe value of IJ vhere section IS beginsThe number of the first panel at span station JThe section number where surface ISF begins.The span station where section IS beginsThe number of span stations in section ISThe number of basic solutions.
The number of panels at span station JThe total number of sectionsThe total number of surfaces
The number of span stations on surface ISFThe number of vortex shell sections
The total number of span stations on all lifting
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NST5
NTB
NTL
NTP
NTS
NTSL
NTV
surfaces.
The number of source span stationsThe number of body panels.The number of vortex shell panels.The total number of panels.
The number of source parameters.The total number of vortex shell span stationsThe number of vortex panels.
LIFTING SURFACE PANELS
CAM(IJ)
COSZ(IJ)
CX(I,IS)DS(J)
DWDX(IJ)
DX(I,IS)ZTA(J)
PA(IJ)
SINZ(IJ)
TWIST(J)
WF(IJ)
WTHK(IJ)
X(KL,IC)
XC(IJ,1)
XC(IJ,2)
E..E(l,J)Xr.E(2,J)XM(I ,IS)
XTE(I,J)
XTZ(2,J)XX(I,IS)
Y(KL,IC)YC(IJ)
Ycc(J)Y0(1,J)YO(2,J)Z(KL,IC)
ZC(IJ)
zcc(a)ZTHK(IJ)
Z0(I,J)
ZO(2,J)
The normal velocity at the control point due toThe cosine of the normal of panel IJ• /c values for the control points of section ISWidth of span station JThe local dv/dx at thickness control point IJDelta (x/c) for the panels of section ISFraction of span running distance for (YCC(J),ZCC(J))
camber.
The area of panel IJThe sine of the normal of panel IJTwist of span station J
The deflections at the control points due to flaps.The local dz/dx at thickness control point IJ• values of the corners of vortex panel KL (4)• value of the centroid for vortex panel IJ• value of the controlpoint for vortex panel IJ• value at the leading left edge of span station J.• value at the leading right edge of span station J.x/c values for the midpoints of panels of section IS• value at the trailing left edge of span station J.• value at the trailing right edge of span station J.x/c values for the panels of section ISy values of the corners of vortex panel KL (2)y value of the control point for vortex panel IJy value at the control point of span station Jy value at the left edge of span station J.y value at the right edge of span station J.z values of the corners of vortex panel KL (2)z value of the control point for vortex panel IJz value at the control point of span station JThe local z/c at thickness control point IJz value at the left edge of span station J.• value at the right edge of span station J.
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BODY PANELS
S(I)
BT(IC,IJ)
BETA2X
IBB(K)
IJB(K)XX(X).Y(K)
XB(IC,KL)XB0(I)
XIN_ET(ZJ)XN(1,IJ)XN(2,IJ)
X.(4)ZN(5)xN(6)
X,(7)X,(S)X,(9)
XP(IC.KL)X0(1,IJ)xo(z.zo)X0(3,IJ)YB(IC._)IP(IC,KL)ZB(IC,KL)ZP(ZC,_L)
SQRT(DY(1)**2 + BETA2*DX(I)**2) I = 1,4
tans**2 + beta2 for each side of body panel IO
= 1. velocity boundary conditions.= 1.- Hach**2 mass fluz boundary conditions.
The body station number of the first station in bodysection K.
The panel number of the first body panel in section K.The number of • body stations for body section KThe number of panels at each body station (half) for
body section K.
• values of the corners of body panel KL (4)The • value of the center of body station I.mass flow coefficient of body panel IJ; 0. = impermeable• - component of body panel IJ normaly - component of body panel IJ normalz - component of body panel IJ normal
x.(1) / z_(8)XN(2) / SQRT(XN(2)**2 +XN(3)**2)XN(3) / SQRT(XN(2)**2 +XN(3)**2)
SQRT(XN(2)**2 + XN(3)**2) / XN(8)
SORT(beta2*XN(1)**2 + XN(2)**2 + XN(3)**2)
SQRT( DI(1)**2 + XN(2)**2 + XN(3)**2} =
• values of the corners of body panel KL (4)
• value of the centroid of body panel IJ
y value of the centroid of body panel IJ
z value of the centroid of body panel IJ
• values of the corners of body panel KL (4)
• values of the corners of body panel KL (4)• values of the corners of body panel KL (4)
• values of the corners of body panel KL (4)
2. * area
planar
planar
planar
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COMPUTED VARIABLES
Cp(IJ,K) The delta-Cp across panel IJ from basic solution K.
UK(IJ,K)VK(IJ,K}WK(IJ,K)PK(IJ,K)
The control point upper surface • velocity.The control point upper surface binormal velocity.The control point upper surface normal velocity.The control point upper surface velocity potential.
US(IJ)
VS(IJ)
WS(IJ)
PS(IJ)
The thickness induced upper surface • velocity.The thickness induced upper surface binormal velocity.The thickness induced upper surface normal velocity.The thickness induced upper surface velocity potential.
The following variables are required for 2nd order solutions only.
WSX(IJ) d/dx of WS(IJ) source normal velocity.WCX(IJ) d/dx of WK(IJ,2) camber normal velocity.
KKX The number of 2nd order boundary condition solutions.
UBE(IJ,K)UBO(IJ,K)
= 6 if there is no body.= 9 if there is a body.
The even symmetry u velocities due to 2nd order b.c.The odd symmetry u velocities due to 2nd order b.c.
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!
!
i SUBPROGRAM_S.
I A * signifies the subroutlne is from program WBODY
i SUBROUTINE FUNCTION
MAIN Sets the main array size for the program.
ABMAIN Reads input data, computes geometry, and sets array
sizes.ACMAIN Controls the flow of the program.
AER02 * Controls the program flow for the computation of Ist
i or 2nd order pressures and loads.AERO2B * Computes the u velocities due to 2nd order boundaryConditions.
AERO2P * Computes the 1st or 2nd order pressures from the
I induced velocities.AERO2V * Computes the odd and even symmetry velocities induced
by the basic solutions.
i AMINMX * Finds the largest, smallest or largest absolute valueof the elements of an array.
BAINFL * Computes normal velocities after the Cp's or normal
i velocities on specified span stations are set equal tozero.
BODYGM * Computes and prints body geometry from input data.
BODYMX Performs calculations for optimizations with paneled
I bodies.BODYRD * Reads body panel geometry from APAS dataset.BODYW * Finds the intersection of the body and the aerodynmaic
i surfaces.CAMBER Computes camber from input.CLCM * Computes the lift drag and moment characteristics of
aerodynamic surfaces.
i CLOPT Controls a Cp optimization.flow of calculation for
CNSTRn Calculates the Cp constraint functions for a Cpoptimization.
I CP2 Adds second order terms to the Cp's for an twistoptmization using second order pressures.
DECRD1 * Reads the input data.
I DISPLY * Prints arrays of characteristics at the panel controlpoints of the aerodynmaic surfaces.DMXI_VE * Moves the elements of a double precision array.DSSPLY * Prints arrays of characteristics at the panel corner
I points of the aerodynamic surfaces.ENDREC * Used with errset to check for APAS type influence
matrix.
!
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FIELD *
FXDX3 *
FLXSOL
FXROLL
GEOM *
GEOMR
GAUSSHSHLDR *
INTRP3 *
INTRP4 *
INTRPX *
INTRPY *
MATRXF *
MATRXT *
MAXCHK
MTXADD *
MTXMLT *
MTXMVE *
NOTZRO *
ORTHO *
PHIXY *
PLOT *
POLAR *
PTNFRM
QIJT
QUADMX
REGION
SQFIT *
Computes first order field point properties usingpreviously calculated influence coefficients andthe first order solution.
Integrates data using a third order curve fit throughthe nearest four points.
Directs the calculation for the add load and basic loadand obtains the deflections due to inertia forces.
Prints rolling moment coefficients for rolling momentflap optimizations on flexible configurations.
Computes the geometric characteristics of theaerodynamic surfaces and panels.
Reads input geometry from geometry dataset.Solves simultaneous equations using Gaussian elimination.Solves sets of linear simultaneous equations in a
least square sense.Interpolates or differentiates data using a third order
curve fit through the nearest four points.Interpolates or differentiates data using a third or
fourth order curve fit through the nearest four points.Interpolates or differentiates properties chordwise on an
aerodynamic surface using subroutine IXTRP4.Interpolates or differentiates properties spanwise on an
aerodynamic surface using subroutine INTRP4.Displays arrays of data.Displays arrays of data.Checks for variable which exceed maximum value.
Adds multiples of two arrays.Multiplies two arrays.Moves the elements of an array.Checks to see if an array has any nonzero elements.Solves sets of linear simultaneous equations using the
method of succesive orthogonalizations. Aquasi-inverse matrix may be computed or, ifpreviously computed, used to perform the solution.
Calculates v (binormal) velocities on aerodynamicsurfaces from phi (for 2nd order solution).
Writes geometry and aerodynamic data on a disk unit
for computer graphics.Calculates first order drag polar and aerodynamic
cofficients.
Calculates Cp's from the constraint function coefficientsin a Cp optimization.
Computes drag coefficients for thickness drag in anoptimization using 2nd order Cp's.
Uses the method of Lagrange multipliers to maximize aquadratic form subject to a set of constraint equations.
Calculates hinge moment coefficients.Calculates leading edge suction using a least square
Curve fit of Cp to cot(x/c).
142
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TRANSQ
TW0PT
TW0PTP
TWOPTQ
TWST
_/TXDRG *
VTXDR2 *
_TXMAX *
VTXLFT *
WCXSTR
WlXTRP *
file
5689
1011121314
Used fortrim polar calculations.
Performs the transformation used for spanwise constraintson twist optimization constraint functions.
Controls flow of calculation for a twist optimization.Calculates panel Cp's which result from a twist
optimization.Calls QUADMX and checks for optimization of quadratic
form.
Separates twist and camber from normal velocities atpanel control points and computes camber z/c's.
Calculates vortex drag in the trefftz plane.Calculates coefficients for vtxdrg.Subroutine used for optimization with vortex lift.Calculates vortex lift from leading edge suction.Calculates the camber constraint functions for a twist
optimization (l_X > 1).Interplolates velocities to various points on panels.
use
input dataPrint outputscratch filescratch filescratch file
storage of influence matricies and quasi-inverseplot output fileAPAS geometry input filefile for output of trim variation calculation
!143
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FLOW DIAGRAM
**********
* AAMAIN ***********
IV
********** **********
* ABMAIN *<==>I<==>* DECRDI *********** J **********
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<==>* ARRAY ***********
Program OPT
V********** **********
* ACMAIN *<==>I<==>* GEOMR *
********** [ **********
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J **********
<==>* GEOM ***********
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J **********
<==>* CAMBER ***********
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ITWIST OPTIMI ZATION
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<==>* REGION ***********
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<==>* FXROLL ***********
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<==>* TWOPTQ *<=>**********
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<=>* TRANSQ ************
***********
<=>* QUADMX ************
V***********
• WBSOL ************
V***********
• ORTHO ************
********** ***********
<==>* TWOPTP *<=====>* MAXCHK *********** ***********
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ORIGINAL PAGE _SOF POOR QUALITY
IV
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Cp OPTI MI ZATI ON
********** **********
<==>* CLOPT *<==>I<==> * WINTRP *
• ********* I **********
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<==>* QUADMX *<==>* _SOL *
• ********* ***********
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<==>* TRIM ***********
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* ORTHO ************
ANALYSIS
********.. ********** ***********
<_...... > <==>* BAINFL *<....... >* QUADMX *<==>* WBSOL *********** ********** ***********
V
**********
<==>* TWST ***********
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<==>* REGION ***********
V********** **********
<==>* _TXDR6 *<=======>* VTXDR2 *********** **********
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<ffiffi>*POLAR ***********
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TEST CASE
Results for the aspect ratio 2.5, sixty-three degree leading edgesweep trapezoidal wing of figure 4 are presented in this section. TheI0 X I0 uniform chordwise and spanwise aerodynamic paneling used foroptimization is indicated and is typical of planform graphics output ofthe program.
A M = 2.0, CL = 0.I0 tri,Bed second order minimum drag due to liftcase of figure 5 is presented in the remainder of this section. A rootchord twist constraint was imposed to remove the small disturbance apexsingularity common to subsonic leading edge problems. This result iscompared to first and second order optima as a function of the
longitudinal stability parameter dCm/dCl or alternatively the pitchingmoment at zero lift, Cmo. The supersonic nonlinear small disturbance
minimum drag results are the first published to the knowledge of theauthor. The best second order result for the present problem is 6%
lower than first order optimization and occurs at approximately twicethe stability level. The first and second order lifting efficiency of aflat plate of the same planform is shown for comparison purposes. Theimpact of twist and camber for the subsonic leading edge case underconsideration is substantial. Additional results are presented inreference 3.
Test case input is presented on pages 151 and 152. Detailed
program output for this case is presented on pages 153 through 202Typical aerodynamic data graphics output is presented on pages 205through Z16.
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_=2.5- .23
" 83 °
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Fisure 4. Three Dimensional Second Order OptimumModel Problem
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ORI,"z-I[',_'_ _r _..... _::_
OF POOR QUALI¥¥
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Figure 5. First and Second Order Opti_ for a
Subsonic Edge Condition
M = Z.O, CL = 0.i
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TEST CASE INPUT
C63-45 OPT T/C = 0.050
C63-45 THICK OPT T/C = 0.050
63-45 OPT T/C = 0.050FREE FORMAT
C < 0. IF NO JETFLAP
C
C **
C
C
C*
C
C
C
C
C
C
C
C
AT M = 4.0 IST NO CONSTRAINT XCG = FREE
AT M = 4.0 2ND TWIST, NF=3.3 XCG = 12.0
AT M = 4.0 2ND TWIST, NF=3.3 XCG = 12.0
C
C
2-I,
CL-TOTAL
4 0.1000# CHORD-CONST F'S < 0. - NONE
53.3
MATRICIES FROM WBODY, INCLUDING INVERSE (-3.0 INCLUDES THICKNESS)7-3.5
2ND ORDER ANALYSIS < 0.
2ND ORDER OPT =,< -I. ; < -2. FOR THKDRG8-1.00
ADD LOAD OPT, = 0. IF < - 2., GEOMETRY ONLY WILL BE PRINTED.12 1.0
12 0.0
IF > 0., THE NL_BER OF CONSTRAINT EQUATIONS PRINTED.14-100.100
CAMBER PRINTOUT = 3.0
15 3.0
GEOME.TRYPRINTOUT = 2.016 0.
C ** TWIST OPTIMIZATION IF > 0.
17 I.
C MOST PRINTOUT = 2.0, NO UPPER LOWER CP < 0.19-I.
C > 0. FOR A PLOT DATASET
20 2.0C* EXTEND FIRST SURFACE TO CENTERLINE
C 26 1.2
C CAMBER SPECIFICATION ( < 0. FOR NO CAMBER INPUT )29-I.
C* XCG
32 12.000
NO XCG CONSTRAINT
32 -99999.
MACH NUMBER
35 4.00
CONTROL PT FOR DRAG CALCULATION (& OPTIMIZATION IF < 0.)36 0.875
CONTROL PT FOR SUCTION CALCULATION < 0 FOR CURVE FIT PRINTOUT37 0.250
151
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CCCCC
C
C*CC
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CAVG SREF SPAN
41 0.0 160.0 0.0
ODD AND EVEN SYMMETRY U,V,W45 334.
ODD AND EVEN SYMMETRY D/DX U,V,W46 334.
ANGLE OF ATTACK
51 0.0 -99.00 99.00CAM-THK
61 11.22 12.22 12.02FIXED SPAN STATIONS
81 1. 2.CAMBER CONSTRAINTS
151 I.E-08
153 0.0 0.0
155 0.0 0.0
0.0
12.02
TWI ST351 0.0 -1.50 0.0 0.0 0.0951 0.0 1.0 1.0 0.0 0.0
10.1-1.
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P,,E_.,REHCF,S
Clever, W. C., Halmuth, H.D., and Shankar, V., WFormulation ofAerodynamic Prediction Techniques for Hypersonic ConfigurationDesign," NhSA CR-158994, February 1979.
Clever, V. C., and Shankar, V., WAerodynamic Prediction Techniquesfor Supersonic/Hypersonic Configuration Design," HkSA CR-165651,March 1981.
Clever, W. C., and Shankar, V., "Honlinear Potential _nalysisTechniques for Supersonic/Hypersonic Configuration 1)esign_ TM NA_qACR-166078, March 1983.
Shankar, V., and Clever, W. C., "Honlinear Potential AualysisTechniques for Supersonic/Hypersonic Aerodynamic Design, w NASAG1-172299, March 1984.
Bonnet, E., Clever, W. C., and Dunn, K., "Aerodynamic Preliminary" Part I - Theory, N_LqACR-165627, April 1981Analysis System II,
Divan, P., "Aerodlnamic Preliminary knalysis System II,"Part II - User's Manual, NASA CR-165628, April 1981.
217
1. Rmort No. I 2. Gov_nmmt AccIiOn No. 3. Recipimt'$Cs_i_ No:
NASA CR-172342 I4. Title and _btiUe _ Report Dote
SUPERSONIC SECOND ORDER ANALYSIS AND OPTIMIZATION August 3O, 1984
PROGRAM USER'S MANUAL e.p,,4ormi.9o,g..iz.,i_
7. Author|t)
W. C. Clever
9. I_forming OrganizationName and AddraIRockwell InternationalP. O. Box 92098
Los Angeles, California 90009
12. _i_.I Agency Name and Addrm=National Aeronautics and Space Administration
Langley Research CenterHampton, Virginia 25665
8. I_rt_min90rg_ization R_ No.
10. Work Unit No.
1I. Contract or Grant No.
NASI-15820
13. Type of Report and Pwiod Co_KI
Contractor report
14. SponsoringAgency Code
IS. SUl_llmimuv Noun
Technical monitors: Noel A. Talcott, Jr. and Kenneth M. Jones
16. Al:_tract
Approximate nonlinear inviscid theoretical techniques for predicting aerodynamiccharacteristics and surface pressures for relatively slender vehicles at
supersonic and moderate hypersonic speeds were developed. Emphasis was placed
on approaches that _uld be responsive to conceptual configuration designlevel of effort. Second order small disturbance theory was utilized to
meet this objective.
Numerical codes were developed for analysis and design of relatively general
three dimensional geometries. Results from the computations indicate good
agreement with experimental results for a variety of wing, body, and wing-body shapes. Case computational times of one minute on a CDC i76 are typical
for practical aircraft arrangements.
17. Key Worm (SugglstlM by Authorls))
Aerodynamic TheoryPotential AnalysisSupersonic/Hypersonic
18. Distribution Statw_nt
19. Security OaWf. (of thisrel_rt)
+ Unclassified20. SecurityClauif. (of thi! plgl)
Unclass ifled21. No. of Pages
21"722. Price
III
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