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NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER
4"
rc/J
ífiui
COMPREHENSIVE EVALUATION OF SIX THIN-WING LIFTING-SURFACE COMPUTER PROGRAMS
by Henry T. Wang
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SHIP PERFORMANCE DEPARTMENT RESEARCH AND DEVELOPMENT REPORT
June 1974 Report 4333
iEF
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4. TITLE (and Sthtit)
COMPREHENSIVE EVALUATION OF SIX THIN-WING LIFTING-SURFACE COMPUTER PROGRAMS
5. TYPE OF REPORT & PERIOD COVERED
6. PERFORMING ORG. REPORT NUMBER
7. AUTHOR(s)
Henry T. Wang
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Naval Ship Research and Development Center Bethesda. Maryland 20034
IO. PROGRAM ELEMENT. PROJECT, TASK
Suhproject S4606, Task 1 703
Work Unit 1-1 1 53-003
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June 1974
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103
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Naval Ship Systems Command
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Thin Wing Pressure Distributions
Lifting Surface Programs Steady Subsonic Flow Hydrofoils
20. ABSTRACT (Continue on rever.. aid. if neceeaary id Identity by block number)
A comprehensive evaluation is made of six computer programs that analyze thin wing lifting surfaces in steady subsonic flow. The programs are compared with one another and with available experimental data for a total of 62 planform cases, 18 camber cases, and two flap cases. For the planform cases, four of the programs were generally in good agreement with each other and with experimental results, one program showed moderate differences.
UNCLASSI FI ED
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20. - Continued
and the remaining program showed large differences. The agreement between predicted and experimental results was erratic for the camber cases and poor for the flap cases.
TABLE OF CONTENTS Page ABSTRACT ADMINISTRATIVE INFORMATION INTRODUCTION I DESCRIPTION OF PROGRAMS 3 MARGASON-LAMAR PROGRAM 4 LOPEZ-SHEN PROGRAM 4 TULINIUS PROGRAM 5 BANDLER PROGRAM 7 LAMAR PROGRAM 8 WAGNER PROGRAM II
COMPARISON OF THE PROGRAMS
.13 PLANFORM CASES
13
Lift Coefficient CL
13
Pitching Moment Center of Pressure PMCP 14
Bending Moment Center of Pressure BMCP 14
Far-Field Induced Drag Coefficient CDI 15
Near-Field Induced Drag Coefficient CDII
15
Summary
16
CAMBER CASES
16
BMCP, CDI, and CDII
18
CL and PMCP
18
COMPUTER RUNNING TIMES
20
COMPARISON WITH EXPERIMENTAL RESULTS 22
PLANFORM CASES
22
Overall Coefficients
23
Spanwise Distributions 24
Chordwise Distributions of Pressure Difference Coefficient 26
CAMBER CASES 27 FLAP CASES 27 CONCLUSIONS 28
ACKNOWLEDGMENTS 30
APPENDIX DISCUSSION OF NEAR- AND FAR-FIELD INDUCED
DRAG COEFFICIENTS 87
REFERENCES 89
LIST OF FIGURES
i Relation of Wing to Vortex Sheet in x-y Plane 31
2 Chordwise Elementary Vortex Distribution 31
3 Plan and Front Views of Experimental Wing-Body Combination 32
4 Plan View of Experimental Planforms 33
5 - Piercy Symmetrical Section 34
6 Experimental Flap Configurations 34
7 Spanwise Distribution of Loading Coefficient for the Wick Experiment
(A = 56.3, r 0, A 2) 35
8 Spanwise Distribution of Loading Coefficient for the Smith-Reed
Experiment (A 56.3, r 0, A = 2) 36
9 Spanwise Distribution of Local Center of Pressure for the Wick Experiment
(A 56.3, r 0, A = 2) 37
I O Spanwise Distribution of Loading Coefficient for the Faikner-Lehrian
Experiment (A = 45, T = I, A = 3) 38
11 Spanwise Distribution of the Local Center of Pressure for the
Faikner-Lehrian Experiment (A 45, r = 1, A 3) 39
12 Spanwise Distribution of Loading Coefficient for the Graham Experiment
(A = 45, r = 0.45, A = 8) 40
1 3 Spanwise Distribution of Local Center of Pressure for the Graham
Experiment (A = 45, r 0.45, A = 8) 41
14 Spanwise Distribution of Loading Coefficient for the A = 0, r = 1,
A = 5 Planform 42
15 Spanwise Distribution of Local Center of Pressure for the A 0, r = 1,
A 5 Planform 43
Page
16 - Chordwise Distribution of Pressure Difference Coefficient for the Smith-Reed
Experiment at i = 0.25 (A = 56.3, r = 0, A = 2)
44
1 7 - Chordwise Distribution of Pressure Difference Coefficient for the Smith-Reed
Experiment at î = 0.6 (A = 56.3,
r
= 0, A = 2) 4518 - Chordwise Distribution of Pressure Difference Coefficient for the Wick
Experiment at 17 = 0.15 (A = 56.3,
r
= 0, A = 2)46
19 - Chordwise Distribution of Pressure Difference Coefficient for the Wick
Experiment at 17 = 0.80 (A = 56.3,
r
= 0, A = 2)47
20 - Chordwise Distribution of Pressure Difference Coefficient for the
Faikner-Lehrian Experiment at = O (A = 45,
r
= 1, A = 3)48
21 - Chordwise Distribution of Pressure Difference Coefficient for the
Faikner-Lehrian Experiment at 17 = 0.50 (A 45,
r
= 1, A = 3) 4922 - Chordwise Distribution of Pressure Difference Coefficient for the Graham
Experiment at 77 = 0.10 (A = 45,
r
= 0.45, A = 8) 5023 - Chordwise Distribution of Pressure Difference Coefficient for the Graham
Experiment at ? = 0.90 (A = 45, r = 0.45, A = 8) 51
LIST OF TABLES
i - Number and Location of Control Points
2 - Variation of Overall Coefficients with Number of Spanwise Stations M as
Predicted by the Lamar Program
3 - Computer Program Predictions of Overall Coefficients for 60
Systematic Planform Cases
4 - Computer Program Predictions of Overall Coefficients for Nine NACA
Meanlines (A = 0,
r
1, A = 5 Planform)5 - Computer Program and Experimental Overall Coefficients for Nine
Designated Meanlines (A = 0,
r
= 1, A = 6 Planform)6 - Slope Distribution of Designated Camber Lines
7 - Running Times for Computer Programs
8 - Summary of Experimental Conditions
9 - Experimental Overall Coefficients for Five of the Systematic
Planform Cases 52 53 54 74 77 80 81 82 83
Page
10 - Computer Program and Experimental Overall Coefficients for A 56.3,
r = 0, A = 2 Planform, a = 10.4 Degrees 83
11 - Computer Program and Experimental Overall Coefficients for A 56.3,
r = O, A = 2 Wing-Body Planform, a = 8 Degrees 84
12 - Computer Program and Experimental Overall Coefficients for A = 45,
r = 0.45, A = 8 Planform, a 4.7 Degrees 84
13 - Computer Program and Experimental Overall Coefficients for Constant
Percent Chord Flap Deflected 5 Degrees 85
14 - Computer Program and Experimental Overall Coefficients for Constant
NOTATI ON
A Aspect ratio
Amn Coefficients of the series for vorticity
BMCP Bending moment center of pressure measured from the wing root and based
on the semispan b/2
b Total span
CDI Induced drag coefficient based on the spanwise distribution of circulation;
also referred to as drag based on the Munk theory or far-field drag
CDII Induced drag coefficient based on a direct integration of the pressure acting
on the wing; also referred to as drag based on thrust or near-field drag
CL Lift coefficient
C Local lift coefficient
Cr Root chord
c Local chord
Average chord
EVD Elementary vortex distribution
f(Ø) Birnbaum chordwise loading function
hrn (ti) Spanwise loading function
K Kernel function
L.E. Leading edge
M Number of spanwise control points on a semispan
Ma Mach number
Mr [(1 to 1.25) (ß A) N + 11/2
M2 Number of spanwise control points on the total span
N Number of chordwise control points
PMCP Pitching moment center of pressure measured from the leading edge of the
wing root and based on the root chord
Projection of the wing in the xy-plane, Figure 1
T Thickness to chord ratio
T.E. Trailing edge
Time for nine camber cases/time for one planform case
U Free-stream velocity
w Downwash
x Coordinate in the direction of the free-stream velocity, Figure 1
Xe x-value of the local leading edge
y Spanwise coordinate, Figure 1
z Vertical coordinate, Figure 1
a Angle of attack, Figure I
¡3 Prandtl-Glauert correction for compressibility
\/l Ma2
'y Vorticity
Pressure difference coefficient
Pressure difference between the upper and lower surfaces of the wing
5 Spanwise width of the integration region around a control point, used in the
Wagner Program
y/(b/2)
O Spanwise angular coordinate = cos
()
A Sweep of the quarter-chord line
(XoeXe)/C
e-value of the local center of pressure
-value at which the slope of the meanline is zero
t-value of the control point nearest to the leading edge
p Fluid density
r Taper ratio = tip chord/root chord
Chordwise angular coordinate = cos
[
ABSTRACT
A comprehensive evaluation is made of six computer programs that analyze
thin wing lifting surfaces in steady subsonic flow. The programs are compared
with one another and with available experimental data for a total of 62
plan-form cases, I 8camber cases, and two flap cases. For the planplan-form cases, four
of the programs were generally in good agreement with each other and with
ex-perimental results, one program showed moderate differences, and the remaining program showed large differences. The agreement between predicted and experi-mental results was erratic for the camber cases and poor for the flap cases.
ADMINISTRATIVE INFORMATION
This work was authorized and funded under the Hydrofoil Development Program
ad-ministered by the Systems Development Office of the Naval Ship Research and Development Center (NSRDC) under Naval Ship Systems Command Subproject S4606, Task 1703,
Work Unit 1-1 153-003.
INTRODUCTION
The present report contains a comprehensive evaluation of six different computer
pro-grams that have been developed in recent years by organizations in North America to analyze
lifting surfaces in steady subsonic flow. The six programs and the organizations responsible
for their development are as follows:
Margason-Lamar Vortex Lattice Program (NASA Langley)1 Lopez-Shen Elementary Vortex Distribution Program
(McDonnell Douglas Corporation)2 Tulinius Program (Rockwell International)3
'Margason, R.J. and i.E. Lamar, "Vortex-Lattice FORTRAN Program for Estimating Subsonic Aerodynamic Character-istics of Complex Planforms," Langley Research Center, NASA Technical Note D-6142 (Feb 1971). A complete listing of references ii, given on page 89.
2Lopez, M.L. and C.C. Shen, "Recent Developments in Jet Flap Theory and Its Application to STOLAerodynamic
Analysis," Paper 71-578, AIAA 4th Fluid and Plasma Dynamics Conference, Palo Alto, California (Jun 1971).
3Tulinius, J., "Theoretical Prediction of Wing-Fuselage Aerodynamic Characteristics at Subsonic Speeds," NorthAmerican
Bandler Program (Engineering Research Associates, Canada)4'5
Lamar Modified Muithopp Program (NASA Langley)6
Wagner Program (NASA Ames)7
The first five of these programs were among 1 5 evaluated in a previousreport8 which
compared the loadings predicted by the programs with each other and with experimental
results for two planform cases. The results from the five pertinent programs were in fairly
good agreement with each other and with experimental results. The Wagner program has
been added here because unlike the other five, it allows control points at the leading and
trailing edges of the wing.
The six programs considered in the present study represent six different approaches to solving the lifting surface integral equation. Each was run for a total of 62 different
plan-form cases and two flap cases. For two given rectangular planplan-forms, five of the programs
(except for the Lamar program) were run for 18 different camber lines. All but two of the 62 planforms represented a systematic variation of aspect ratio from I to 1 2, sweep of the quarter-chord line from O to 45 degrees, and taper ratio from 1 to 0. In the remainder of the report, these 60 planforms will be referred to as the systematic planforms. The above
range of planform and camber cases covers most cases of interest in subsonic flow. Thus. an
evaluation over this range will give a good indication of the usefulness of these programs for
such applications.
Five of the programs were run on the CDC 6700 computer currently being used at NSRDC. These were modified to make them compatible with the computer and to standardize their output. A substantial portion of the effort of the present study was in-volved with these modifications. The Lopez-Shen program was run on an IBM 370/165 by
the McDonnell Douglas Corporation under contract to NSRDC. The results are reported in
Reference 9.
4Richardson, J.R., "Pressure Distribution on a Hydrofoil of Finite Span neat the Free Surface," Engineering Research Associates Report 53/3 (Jun 1966).
5Bandler, P.A., "A Program to Calculate the Pressure Distribution on a Hydrofoil of Finite Span near the Free Surface," Engineering Research Associates Report 53/4 (Jun 1966).
6Lamar, LE., "A Modified Muithopp Approach for Predicting Lifting Pressures and Camber Shape for Composite
Plan-forms in Subsonic Flow," Langley Research Center NASA Technical Note D-4427 (Jul 1968).
7Wagner, S., "On the Singuhirity Method of Subsonic Lifting-Surface Theory," J. of Aircraft, Vol. 6, No. 6, pp. 549-558
(NovDec 1969).
8Langan, T.J. and H.T. Wang, "Evaluation of Lifting-Surface Programs for Computing the Pressure Distribution on Planar Foils in Steady Motion," NSRDC Report 4021 (May 1973).
The results from the six programs are presented in the form of tables for the overall
coefficients and graphs for the spanwise and chordwise distributions of the loading. These
results are first compared with one another and then with available experimental data for
seven of the planform cases, nine of the camber cases, and the two flap cases.
DESCRIPTION OF PROGRAMS
The programs are all based on the assumption that the flow is steady, irrotational, and attached. When compressibility effects are included, they are represented by application of the Prandtl-Glauert similarity rule, that is, by a contraction of the x-coordinate, where the x-coordinate is in the direction of the free-stream velocity; see Figure 1. The calculations
presented in the present report were obtained by using linearized thin wing theory for a single
wing. Such effects as the presence of the ground or a free surface, dihedral, and jet flaps are not considered. The problem described above then reduces to the solution of the following well-known lifting-surface integral equation
w(x,y) =
JJ
p(X,w) K(xX, yw, Ma)dX dw (1)sw
where w downwash and is given by the product of the free-stream velocity with the
local streamwise slope of the mean camber surface
y spanwise coordinate (Figure 1)
p = fluid density
U = free-stream velocity
S,
= projection of the wing in the xy-plane (Figure 1)= the unknown pressure difference between the upper and lower surfaces ofthe
wing
K = kernel function which contains a strong singularity as y - w
Ma = Mach number
The present section describes in some detail the approaches used by the various programs
to solve the above singular integral equation. The particular way these programs were used to
generate the results presented herein is described, and a brief mention is made of their
capabilities beyond the thin wing-alone problem considered in the present report. Detailed
MARGASON-LAMAR PROGRAM
This is a straightforward vortex lattice program1 and is representative of a number of
similar programs currently in use. As shown in Figure 1, the wing is divided into a number
of panels. Each panel is replaced by a horseshoe vortex which has a vortex filament across
the quarter-chord of the panel and two streamwise filaments that trail to infinity. The flow tangency condition is satisfied at control points which are located at the midspan,
three-quarter chord point of each panel. This leads to a system of linear simultaneous equations
for the unknown vortex strengths. Once the vortex strengths are known, the lift on each panel can be obtained by using the Kutta-Joukowski theorem. The other aerodynamic data
can be obtained from the lift.
For planform cases, experience with this program1 ,8 shows that for a given number of
chordwise panels, most of the overall aerodynamic coefficients converge to at least two and
possibly three figures for 30 or more evenly spaced spanwise panels. The results also
indi-cate that the overall aerodynamic coefficients for planform cases usually show little change
when the number of chordwise panels is increased beyond four. Thus, the number of
chordwise and spanwise rows was fixed at 4 and 30, respectively, for all the results presented
in the present study. It may be noted that the resulting number of panels (1 20) isthe
maximum allowed by the program.
It may be reasonably pointed out that four chordwise control points are not sufficient for many camber cases. However, one purpose in studying the camber cases was to
determine the effect of choice of location and number of chordwise control points on the
predicted overall aerodynamic coefficients of cambered wings. Since the number and/or
location of control points used to calculate the planform cases differed among the various
programs, these same choices of control points were also used to calculate the camber and
flap cases for all programs except those of Tulinius and Lopez-Shen. Table I shows the number and location of the chordwise and spanwise control points used by each program,
including the various sets of locations for the Tulinius and Lopez-Shen programs.
In addition to the capabilities needed to calculate the results shown in the present study, this program can deal with the case of a wing with dihedral and the case of a wing in the presence of another wing or tail.
LOPEZ-SHEN PROGRAM
The method of elementary vortex distributions developed by Lopez and Shen2 is
referred to as the EVD method. The wing planform is approximated by a number of rec-lines span
the entire chord of the approximate planform. A vortex distribution is placed over each pair \of rectangles which are adjacent in the chordwise direction. On each pair of rectangles, the
distribution is uniform in the spanwise direction and triangular in the chordwise direction with a peak at the common side; see Figure 2. It is this linear variation of the vorticity in the chordwise direction which distinguishes this program from classical vortex lattice programs such as that of Margason and Lamar. At the leading edge and flap hinges, one may choose
to use leading edge EVD's arid hinge EVD's, respectively. The flow tangency condition applied at the center of each rectangular panel leads to a system of simultaneous linear equations for the unknown peak (or average) values of the elementary vortex strengths. Once
these vortex strengths are available, all the other aerodynamic data can be obtained.
Thirteen spanwise panels were used to generate all of the results for this program. For the planform cases with taper ratios of 1.0 and 0.5, II chordwise panels were used on the entire wing. For the zero taper ratio planform cases, 11 chordwise panels were used for
80 percent of the semispan closest to the wing root and 6 chordwise panels for 20 percent of the semispan next to the wing tip. This chordwise arrangement for the zero taper ratio cases (the number of chordwise panels is decreased near the wing tip where the chordwise dimensions are small) was found to improve the convergence of the linear algebraic system.* Table I shows the special chordwise panel arrangements used in the flap cases. Note that for the camber cases (Table 1), a second run was made with 20 chordwise panels to obtain camber results with maximum accuracy.
In addition to its capability for solving the thin wing problem considered in the present report, this program can also handle wings with a jet sheet of varying strength issuing along the trailing edge.
TULINIUS PROGRAM
This program3 is a hybrid of the vortex lattice approach represented by the Margason-Lamar program described above and the three functional expansion approaches described later. As in the vortex lattice approach, the wing is divided into a discrete number of panels with a horseshoe vortex placed in each panel. However, whereas the strength of each vortex is an
unknown to be solved for in the vortex lattice approach, the strengths of the vorticesare
related in this program by a series of functions with unknown coefficients which are to be solved for. This series for the vorticity takes the form
*
=
where 'Y = vorticity
= chordwise angular coordinate =
cos1
[12(X_XQe)]
Xe x-value of the local leading edge
c = local chord
= y/(b/2)
b = the total span
A = unknown coefficients to be determined
M_PM M
2p 2
hm(1?)=V/17l2 amp ri + amp P(ri) p=1 p=M_PM+lf() = cot (Ø/2)
n 1= sin (nl)
n = 2,3,», N1, N
= P-functions which are used to account for breaks in the wing leading or trailing edge; these functions are defined in Appendix E of Tulinius.3
The f() terms are the well-known Birnbaum chordwise loading functions. The hrn )
terms without the P-functions are simply polynomial perturbations of the basic elliptic
span-wise distribution given
by/l_i12. The
and hm(ri) terms without the P-functions havebeen employed in a number of lifting surface programs which use the functional expansion approach. The P-functions appear to be unique to this program. The above functions may also be supplemented by special flap functions.
The (N)x(M) unknown coefficients Amn are determined by satisfying the flow
tangency condition at a prescribed number of control points equal to or greater than (N)x(M).
Thus the solution is obtained by a least-squares technique. In straightforward functional
ex-pansion approaches, the downwash at the control points is obtained in terms of the Amn by evaluating the singular integral equation, Equation (1). This difficulty is avoided here by
summing the downwash contributions of the individual vortices. The resulting downwash
still contains the coefficients Am since they determine the strengths of the individual
vortices.
In order to generate the results shown in the present report, all runs except one flap run were made with 6chordwise and 6spanwise functions, including a P-function for the wing
root where there is a break for the swept and/or tapered planforms. The vortex panels were
arranged into 20 chordwise and 21 spanwise rows. For all runs, the number of spanwise
control points was set equal to il. The number of chordwise control points was set equal to
9 for the planform cases and 10 for the camber and flap cases.* As shown in Table 1, the locations of the control points are identical for all cases except that the camber and flap cases have one control point closer to the leading edge than the planform cases. This was done in the camber cases to account for any rapid changes of camber slope which might occur near the leading edge.
This program has a number of other capabilities. It can account for wing thickness as
well as the presence of a fuselage.
BANDLER PROGRAM
This program4 represents a functional expansion approach in which the pressure
difference zp is expressed as a double series of functions with unknown coefficients and the
downwash is obtained by directly integrating the singular integral, Equation (I). The pressure
difference p is given by
*
These values were recommended by Mr. Jan R. Tulinius of Rockwell International.
[ cosn+cos(nl)1
j
sin (2 m-1)O (3)where O is the spanwise angular coordinate and is equal to cos1(i); Ø and 77 are defined
after Equation (2).
The integral Equation (I) is solved by using Gaussian integration to evaluate the integral at (M)x(N) control points. Satisfaction of the flow tangency condition at these control points yields a system of (M)x(N) linear algebraic equations for the (M)x(N) unknown coefficients Am0 The chordwise and spanwise locations of the control and integration
M N
p(çb,O)
stations are respectively obtained from equal increments of the chordwise and spanwise angular coordinates Ø and O. It is shown10'11 that these locations yield maximum accuracy for a given number of stations.
At present, this program is limited to a maximum of four chordwise and four spanwise
functions. All of the runs were made by using this maximum number of functions. Since this program was written to analyze hydrofoils, it allows for the effects of free surface and Froude number. Thickness effects are accounted for in a linearized sense, i.e., at
infinite depth, thickness does not affect lift but serves only to determine the pressure
distri-bution over the upper and lower parts of the foil. These pressure distridistri-butions are of interest
in determining the cavitation characteristics of a hydrofoil. More recently, the program has
been expanded to analyze a hydrofoil with pod and strut.12'13
LAMAR PROGRAM
This program is a modified version of the Multhopp subsonic lifting-surface theory.'1
It
is a functional expansion approach which, unlike the Bandler program described above, does not express zp as a double series of functions. Instead ¿p is given by
N
q(rn)
=
n()
(4)n=1
m = l,2,",M1,M
where q(n) represents the unknownvalue of the n th chordal loading function at the m th
dimensionless spanwise station Tm and f() is defined after Equation (2).
The above shows that Lamar and Tulinius useidentical chordwise loading functions. The
series of Equation (4) differs from that of Equation (2) in that the unknowns in the latter
are the coefficients of a double series of functions whereas the unknowns in Equation (4)
10Hsu, P.T., "Flutter of Low Aspect-Ratio Wings Part ICalculation of PressureDistributions for Oscillating Wings of
Arbitrary Planform in Subsonic Flow by the Kernel-Function Method," Massachusetts Institute of Technology, Aeroelastic and Structures Research Laboratory Technical Report 64-1 (Oct 1957).
11Multhopp, H., "Methods for Calculating the Lift Distribution of Wings (Subsonic Lifting-Surface Theory)," Aeronautical Research Council (Great Britain) R. & M. 2884 (Jan 1950).
12Richardson, J.R., "Pressure Distribution on a Hydrofoil with Pod and Strut near the Free Surface," Engineering
Research Associates (Canada) Report 53-5 (Oct 1968).
are the values of the N chordal loading functions f(Ø) at each of M spanwise stations.
Thus, the spanwise functions are never explicitly written. The integral equation, Equation (1),
is solved by evaluating the integral at (M)x(N) control points. Satisfaction of the flow
tangency condition at (M) spanwise x(N) chordwise control points furnishes (M)x(N) linear
algebraic equations for the (M)x(N) unknownsqn(n). As in the Bandler program, the
con-trol points represent equal increments of the chordwise and spanwise angular coordinates.
The principal modifications to the Multhopp approach'1 made in the present program
involve solution of the linear system by matrix inversion rather than iteration, using
Gaussian quadrature rather than a MacLaurin series approach to perform the integration, and
increasing the number of possible chordal functions from two to ten.
The convergence of this program has been studied in some detail6 for unswept wings of
aspect ratios 2 and 7. From this study, it is recommended that the number of spanwise
collocation stations on a wing semispan Mr be related to the number of chordal loading
functions N as follows
(I to l.25)(j3A)N+l
Mr - (5)
2
where ß is the Prandtl-Glauert correction for compressibility and is equal to /l Ma2, and A
is the aspect ratio. For larger values of M, the overall coefficients do not generally differ
appreciably from their respective values at M = Mr
In the present study, the convergence of this program was investigated for a number of
the planform cases considered. It was found for low values of sweep that the above formula
resulted in overall coefficients which appear to have converged. It was observed, however, that in order to achieve converged values of the bending moment center of pressure, M must have a minimum value of approximately 9. This is an important point to note for cases of
low aspect ratio and low values of N where the above formula may predict values of Mr less
than 9. With increasing sweep, the overall coefficients appeared to require increasingly greater values of M in order to reach convergence. For example, consider Table 2 which shows the values of the overall coefficients for different values of spanwise control stations M for four wings with aspect ratio A equal to 5, taper ratior equal to 1, number of chord-wise functions N equal to 3, and values of the quarter-chord sweep A equal to 0, 15, 30, and 45 degrees. For A = 5, N = 3, and Ma = 0, the use of Equation (5) yields a nominal
value for M of 8 to I 0. The overall coefficients appearing in this table are defined as in the
notation. They are used to report overall wing aerodynamic data throughout this report. The significance of computing two values for the induced drag coefficient is discussed in the appendix.
Comparison of results for M = 11 and 15, both of which exceeded the value of Mr for the 0- and 15-degree sweep cases shows that all the overall coefficients have reached
con-vergence. The force coefficients CL, CDI, and CDII have converged to within 0.01. The
centers of pressure PMCP and BMCP have converged to 0.001 of root chord and semispan,
respectively. For the 30- and 45-degree sweep cases, the convergence of coefficients CL,
BMCP, and CDI is similar to that for thelower sweep cases, but the convergence of
coefficients CDII and PMCP is poorer. For the 30-degree sweep case, CDII appears to have
converged to only within 0. 1 whereas PMCP has converged to perhaps 0.003 of root chord.
For the 45-degree sweep case, CDII has not even converged to the first decimal place whereas PMCP has converged to only 0.01 of root chord.
It appears from the above discussion that M values considerably in excess of thosegiven
by Equation (5) are needed to obtain converged values of all the overall coefficients.* Thus
it seems that the capability that allows for a large number of chordwise functions (N can go
up to 10) cannot be utilized for wings with large aspect ratio and/or sweep.
The results presented in the following sections for the plariform cases were obtained by
using for a given value of N a value of M which either was at least 30 percent in excess of
that given by Equation (5)** or, in the case of the runs for A = 12 and N = 3, was near the
program limit of M = 21. Table 3 shows two sets of overall coefficients for the Lamar
program for each of the 60 systematic planform cases. These sets correspond to two
different numbers of chordwise functions. When the various programs are compared in the
following section, the coefficients not enclosed in parentheses are considered to be the
coefficients for the Lamar program. With the exception of the A = 12 runs, this set
corresponds to that with the higher number of chordwise functions. The main reason for
showing the set enclosed in parentheses is to indicate the effect on the overall coefficients
of changing the number of chordwise functions. In addition, the results for N 2, M = 7
for A = I serve to show that seven spanwise stations are still not sufficient to obtain
satisfactory values for BMCP. In the case of A = 1 2, the lower number of chordwise
functions is selected since the M value for the higher number is only equal to Mr It has
ready been shown that it is necessary to choose M greater than Mr for swept wing cases.
The flap and camber cases were run with N = 4 and M = 16. Only two camber runs
were made for the following reasons:
1. Since the program can handle only one camber at a time, a separate run would have had
to be made for each of the 18 camber cases, thus increasing computer costs considerably,
*
In a private communication, Dr. John E. Lamar of NASA Lang'ey indicated that this program ispresently being
modified to give the correct values of M for sweptwings.
**
2. For N = 4, the chordwise control poìnts are identical to the chordwise control points of
the Bandler program.4'5 It is shown later in Table 4 that the two camber results for N 4
obtained by using this program are very similar to the corresponding Bandler camber results.
WAGNER PROGRAM
This program7 is another modified version of the Muithopp theory. As in the Lamar program,6 the pressure difference is given by a series of tl1e form
N
E
q(rn)
(6)n1
m = l,2,",M-1,M
where q(n) is defined after Equation (4) and
2
cos(nl)
+ cos nç= -
. n =1.
sin(nl)
The novel feature of this program is that control points are placed at the leading and
trailing edges. This is not allowed in any of the otherprograms considered in the present
study. In the version described by Wagner,7 the chordwise control points are equally spaced while the spanwise coordinates are spaced by equal increments of the spanwise angular coordinate O. In the version used to generate the results in the present report, the chordwise control points are spaced by equal increments of the chordwise angular coordinate
Other modifications to the Multhopp approach include the use of Gaussian quadrature
in the chordwise direction and spanwise integration stations which are not identical with
the control points. To perform the spanwise intergration for a given control point, the wing is divided into a number of regions. One region consists of a narrow chordwise strip of
width 2 in the spanwise direction around the given control point. The width of this region
is an input to the program.
In order to gain experience with this program, several runs were made for the two
planform cases considered by Langan and Wang.8 Both wings had an aspect ratio of 5. One
wing had a taper ratio of 1 and quarter-chord sweep of 1 5 degrees and the other a taper
*
ratio of 0.5 and no sweep of the quarter-chord. In these runs, the dimensionless width b/2c, the number of chordwise control points N, and the number of spanwise control points
on both semispans M2 were varied. The results were then compared with the previous data.8
It was found that when M2 had an odd value such that control points were placed at
the wing root, most of the overall coefficients differed noticeably from the earlier results.8
This difference was particularly large for the wing with the 1 5-degree sweep. One
character-istic for this wing was that the near- and far-field induced drag coefficients CDI and CDII
differed by over 40 percent. This point is discussed further in the appendix. Since both wing planforms have breaks at the wing root, the above trend seems to support the results
given in Appendix E of Tulinius et al.14 where it is shown that certain integrals in the
Wagner formulation lead to logarithmic infinities of the downwash at regions where there
are breaks. According to Tulinius et al., these singularities are simply dropped by Wagner.
Thus this program should not be used with odd values for M, which give rise to control
points at the wing root.
For even values of M2, a value of b/2c = 0.005 gave reasonable values for the overall aerodynamic coefficients for a range of values of M2 (which is equal to 2M when M2 is even) and N. With one exception, all of the planform, camber, and flap results shown for this
program in the following sections were obtained by using i5b/2c = 0.005, M2 = 16 (i.e.,
M = 8), and N = 5. This corresponds closely to the values of M, = 1 5 and N = 5 used by Wagner.7 However, it must be noted that with M, = 1 5, control points are placed at the
wing root. The one exception was the planform case A = 8, A 0, and r 0. Table 3.14
shows the two sets of values for this case where the Wagner program employed ôb/2c
= 0.005 and 0.003 5. The reason for this is as follows. A comparison of the results for the
Wagner program using tb/2c = 0.005 for A O and r= O given in Tables 3.11 to 3.15 shows
that PMCP increases monotonically with aspect ratio except for the A 8 case.
Further-more, Table 3.14 shows that CDII is only one-half of CDI; for the other aspect ratios, CDI
and CDII differ by only a few percent. This indicated that some other value of b/2c
should be used for this case. Several other values of öb/2c were used, ranging from 0.002 to
0.10. The overall coefficients corresponding to ôb/2c 0.0035 are shown in Table 3.14.
Note that this gives a value for PMCP which is in much closer agreement with the other
pro-grams and also a value ofCDII that is much closer to CDI. When the various programs are
compared in the following section, the coefficients corresponding to the case of öb/2c = 0.0035
are considered to be the values from the Wagner program. To avoid confusion, the coefficients corresponding to Sb/2c = 0.005 are given in parentheses in Table 3.14.
'4Tulinius, J. et al.. "Theoretical Prediction of Airplane Stability Derivatives at Subcritical Speeds," North American Rockwell Report NA-72-803 (1972).
COMPARISON OF THE PROGRAMS
The present section compares the overall coefficients and computer running times of the
various programs for the systematic planform cases and the camber cases. Detailed spanwise
and chorciwise distributions of the loading are compared later when the program results are also compared with experimental results.
PLANFORM CASES
The 60 systematic planform cases for which the overall coefficients CL, PMCP, BMCP, CDI,
and CDII are shown in Table 3 represent a systematic variation of aspect ratio (A 1,3,5,8,
12), taper ratio (r = 1,0.5,0), and quarter-chord sweep (A = 0,15,30,45 degrees). The table
is divided into 60 sections, one for each of the planform cases. The sections are arranged in such a way that aspect ratio is cycled the fastest, the taper ratio is intermediate, and the
quarter-chord sweep is cycled the slowest. Thus, Table 3.1 corresponds to (A = 1,
r
I,A = 0), Table 3.2 corresponds to (A = 3, r = 1, A = 0), Table 3.6 corresponds to (A 1,
r = 0.5, A = 0), Table 3.16 corresponds to (A = 1, r I, A = 15), and so forth. The
pro-grams are first discussed with respect to each overall coefficient and then general conclusions are drawn.
Lift Coefficient CL
The six programs are in fairly good agreement for the A = O cases. The spread between all the programs is typically 2 to 3 percent. The Lopez-Shen program is usually the highest and the Bandler program is often the lowest. The spread between the other four programs is usually within 1 percent.
There is a clear differentiation between the programs for A 1 5 degrees. Except for
A = I (for which the Lopez-Shen program gives the highest value), the Bandler program al-most always gives the highest value, followed by the Lopez-Shen program. The values given by the Bandler program are often more than 5 percent higher than thOse from the Lopez-Shen program. The Lopez-Lopez-Shen value, in turn, is typically 2 percent higher than the highest value of the remaining four programs. The Tulinius program usually gives the lowest value, and this deviation increases with increasing sweep and aspect ratio. For A = 1 2, r = 1, and A = 45 degrees, the Tulinius value is 4 percent lower than any of the other programs.
The spread between the values of the remaining three programs (Margason-Lamar, Wagner, and Lamar) is usually less than I percent and seldom exceeds 2 percent. The values of the Wagner and Lamar programs often agree to three figures.
Pitching Moment Center of Pressure PMCP
For A 0, the typical spread between all six programs is 0.3 to 0.4 percent of root
chord Cr (The Margason-Lamar program usually yields the highest values.) An exception to this trend occurs for A = 1 where the Bandler program yields values which are 0.6 to 1.0 per-cent of C lower than any of the other programs.
For A 1 5 degrees, the Bandler program always yields the lowest values and the
Lopez-Shen program almost always yields the highest values (except for some A = 1 cases). Their deviations from the remaining four programs generally increase with increasing sweep. The largest deviations of the Bandler program from the others for A = 1 5, 30, and
45 degrees are respectively 1.6, 3.5, and 7.8 percent of Cr The Lopez-Shen program yields
values which are typically several tenths of a percent of a C higher than any of the other
programs for A = 1 5 and 30 degrees. At A 45 degrees, this deviation increases to several
percent of root chord.
Of the remaining four programs, the Tulinius values are usually the highest and the Margason-Lamar values the next highest. The difference between these two programs
generally increases with increasing sweep. The Wagner and Lamar values are usually in good agreement with a typical difference of 0.2 to 0.3 percent of Cr
Bending Moment Center of Pressure BMCP
The Lopez-Shen program always gives the highest values. For A 30 degrees, values
are typically 0.5 to 0.7 percent of semispan higher than those given by any of the other
programs. For A = 45 degrees, the deviation may be as high as 1.5 percent of semispan.
The remaining five programs are generally in good agreement for A = 0, with a typical
spread of 0.5 percent of b/2. For A 15 degrees, the Bandler program generally gives the
lowest values (except for some low aspect ratio cases). This deviation, which typically
ranges from 0.5 to 1.0 percent of b/2, generally increases with increasing sweep and aspect
The spread between the values of the remaining four programs usually ranges from 0.3
to 0.6 percent of b/2 for A 30 degrees. For A = 45 degrees, the spread may exceed
LO percent of b/2 with the Tulinius values at the high end and the Wagner values at the low end.
Far-Field Induced Drag Coefficient CDI
The trend for CDI* generally follows the pattern of CL. That is, for A 15 degrees,
the Bandler values are usually the highest and the Lopez-Shen values next. Occasionally, the Bandler program predicts a higher value of CL than the Lopez-Shen program, but the latter predicts a higher value of CDI. As in the case of CL, the Tulinius program predicts the lowest values for high values of the sweep and aspect ratio.
The spread of the values between the remaining three programs usually does not exceed
2 percent. It may be noted that this spread is somewhat greater than that for CL.
Near-Field Induced Drag Coefficient CDII
The CDII* values will be discussed in terms of the agreement between the CDI and CDII values for each program. The significance of this agreement is discussed in the appendix.
The Bandler program has clearly the poorest agreement. The difference between the two values for induced drag usually exceeds 100 percent and in many cases CDII is negative. Next comes the Tulinius program for which the difference is typically 30 percent. Except
for the five cases with A = O degree and r 1, CDII is always greater than CDI for this
program. The difference for the Lamar program is typically about 1 O percent.
The remaining three programs (Margason-Lamar, Lopez-Shen, and Wagner) all have
differences which are typically 5 to 1 0 percent. The Margason-Lamar program gives values of
CDII which are always less than CDI. On the other hand, the Lopez-Shen program gives values of CDII which are almost always (two exceptions) greater than CDI.
Summary
The Bandler program deviates by far the most from the other programs. For each of
the five overall coefficients, it generally shows the greatest deviation. It should be noted
that the Bandler program is limited to a maximum of four chordwise and four spanwise functions and control points. Thus, as shown later, the computer time requirements of this program are substantially less than those of any other program. Accordingly it may be worthwhile to modify the program so that it can handle additional control points and functions.
The Lopez-Shen program is clearly the program with the next greatest deviation. This was first attributed to the fact that the shape of the panels is restricted to rectangular, thus leading to inaccurate representation of the wing planforms with sweep and/or taper. How-ever, Tables 3.1 to 3.5 show that even for the rectangular planforms, this program con-sistently gives the highest value of CL, BMCP, and CDI. Perhaps the explanation is the
location of the control point at the middle of each panel. In the classical vortex lattice
approach, such as the Margason-Lamar program, the control point is placed at the three-quarter chord of each panel. Thus, it would be of interest to investigate the effect on the overall coefficients of varying the chordwise location of the control point within each panel.
Of the remaining four programs, the Tulinius program shows significant deviations from the other three for high values of sweep where there is a sharp break at the root chord in the wing planform. It has been pointed out that this program uses special P-functions to account for breaks in the wing planform. Their presence probably accounts for these deviations at high values of sweep. Comparisons with experimental results, shown later in the present report, will give an indication of the accuracy of using these P-functions.
The results given by the other three programsMargason-Lamar, Wagner, and Lamar-which respectively represent the classical vortex lattice approach and two modifications of the Multhopp approach are generally fairly close.
CAMBER CASES
Tables 4 and 5 present the overall coefficients from five of the programs for a total of 1 8 camber lines for rectangular wings of aspect ratio 5 and 6, respectively. As previously mentioned, the Lamar program was run for only two camber cases and the Lopez-Shen program for two sets of chordwise control points.
The wing with aspect ratio 5 was run for nine NACA meanlines with a Mach number equal to zero. These well-known meanlines are tabulated in Abott and Von Doenhoff,'5 and are not repeated here. The wing with aspect ratio 6 was run for nine meanlines for
which experimental data were available.16'17 In the remainder of the report, these
mean-lines will be referred to as the designated meanmean-lines. Their chordwise slope distributions
were computed from the ordinates of the upper and lower surfaces of the airfoil given in
Norton and Bacon.167 These slope distributions are given in Table 6; the slopes presented
there are tl1e negative of the geometrical slopes. Although these slopes are opposite to the
sign convention used in Abott and Von Doenhoff,'5 they are in accordance with the sign
convention used to enter input data for all the computer programs considered in the present
study. For those programs which could account for compressibility, namely,
Margason-Lamar, Tulinius, and Wagner, the runs were made at a Mach number of 0.053, which
corresponds to one of the Mach numbers used in the experiments.16'17 The Prandtl-Glauert
correction factor ß corresponding to this Mach number is 0.9986, which is very nearly equal
to the incompressible case of ¡3 = 1 .0. Thus, compressibility effects may be neglected here.
Tables 4 and 5 show for each meanline the value of (which is the distance from the
local chord leading edge in terms of the local chord) at which the slope of the meanline is zero. This serves to give an indication of the distribution of the slope changes along the
chord. For example, if is much less than 0.5, then there are rapid changes in slope in the
leading portion of the wing. If is around 0.5, the slope change is more evenly distributed
along the chord. Table 4 shows that the value of for the NACA meanlines ranges from
0.05 to 0.60 while Table 5 shows that is a constant equal to 0.33 for all the designated
meanlines. Table 5 also gives the thickness to chord ratio T for each meanline. The results are presented in the order of increasing values of T.
Table 3.3 indicates that for the rectangular A = 5 wing used for the NACA meanlines,
the spread in the coefficients CL and PMCP for the planform case between the five programs
run for the camber cases is only 2.3 percent and 0.3 percent of C, respectively. Thus, any
larger differences in these two coefficients for the camber cases are mainly due to the error
of representing the various camber lines by the different number and location of chordwise
control points used by the various programs. Similar remarks apply to the A = 6 rectangular wing used for the designated camber lines.
15Abott, I.H. and A.E. Von Doenhoff, "Theory of Wing Sections," Ûover Publications, Inc., New York, N.Y. (1959).
16Norton, F.H. and D.L. Bacon, "Pressure Distribution over Thick AirfoilsModel Tests," Langley Memorial Aeronautical
Laboratory, NACA Report 150 (1922).
17Norton, F.H. and D.L. Bacon, "The Aerodynamic Properties of Thick Airfoils, II," Langley Memorial Aeronautical Laboratory, NACA Report 152 (1922).
Table I presented the number and location of the control points used by each program
to compute the camber cases. The Margason-Lamar program uses four equally spaced
chord-wise control points with a value of 0. 1875 for the location of the control point nearest
to the leading edge. The Bandler program uses four chordwise control points with 0. 11 7.
The Wagner program uses five control points with = O for the NACA meanlines and
= 0.0 125 for the designated meanlines. The reason for using different values of for
the NACA and designated camber cases is the change in sign of the derivative of the slope
near the leading edge of the designated meanlines shown in Table 6. For example, Table 6
shows that at 0, some of the designated meanlines have positive slopes which are not at
all representative of those near the leading edge. Thus, it was felt that theplacement of a
control point right at the leading edge would give misleading results. The Tulinius program
uses 10 chordwise control points with = 0.024 to solve for six chordwise functions. The
Lopez-Shen program uses two sets of chordwise control points: N = 11 and = 0.025,
N = 20 and = 0.015.
The present section compares in some detail the coefficients CL and PMCP of the
various programs. Comparison of the program predictions with the experimental results
shown in Table 5 for the A 6 wing is deferred until later when the program predictions
are compared with various experimental results.
BMCP, CDI, and CDII
The other three coefficients BMCP, CDI, and CDII which are presented in Tables 4 and
5 will not be discussed extensively. Table 3.3 shows that there is a spread of 1.0 percent of
semispan in BMCP between the five programs for the planform case. Tables 4 and 5 indicate
that for camber cases, the general spread in BMCP between the programs is also about
1.0 percent of semispan; the values of BMCP from the various programs are generally in the
same relative order as for the planform case. Thus, the number of chordwise control points
does not seem to have a large effect on BMCP. The coefficients CDI and CDII are
con-siderably smaller than CL. Generally, CDI follows the same pattern as CL. As in the
plan-form cases, there is aconsiderable spread among the values of CDII.
CL and PMCP
Consider first the results for the five NACA meanlines 210, 230, 62, 64, and 66 shown
generally characterized by large negative slopes at the leading edge and then a monotonically
increasing slope to the trailing edge. The slope changes near the leading edge are
character-ized by . The smaller the value of , the greater are the changes in slope near the
leading edge. Thus, the 21 0 meanline has the largest slope changes near the leading edge while the 66 meanline has the most gentle changes near the leading edge. Table 4 shows the not
unexpected result that the spread between the six sets of results generally decreases with
in-creasing . This is because with increasing , there is a decrease in the error incurred by not having points near the leading edge. The Margason-Lamar program, which has the
lead-ing control point furthest from the leadlead-ing edge, generally predicts the highest values of CL
and the lowest values of PMCP. The spread in CL and PMCP among all six sets of results
ranges from a minimum of 1.1 percent and 2.4 percent of Cr for the 66 meanline to a
maximum of 14.4 percent and 14.7 percent of Cr for the 230 meanline. If the
Margason-Lamar results are excluded, the spread in CL ranges from a minimum of 1 .1 percent for
the 66 meanline to a maximum of 6.3 percent for the 210 and 230 meanlines. The
corresponding spread in PMCP ranges from a minimum of 0.7 percent of Cr for the 66
mean-line to a maximum of 8.6 percent of Cr for the 210 meanmean-line.
It is interesting to note that the spread among all the results is less for the 21 0
mean-line, which has the lowest value of than for the 230 meanline, which has a higher value
of . The reason for this is that for very low values of , the region where there are large
changes in slope is very small and hence the error in the overall coefficients incurred by
neglecting this region may also be small. The results of Table 4 suggest that the maximum
error incurred by an insufficient number of chordwise control points occurs at some nonzero
small value of E0 perhaps 0.10.
Now consider the other four NACA meanlines a = 0.2, a 0.5, a = 0.8, and a = 1.0.
Table 4 shows that they are characterized by relatively large values of E ranging from 0.33
to 0.52. The above discussion implies that the spread between the values would be
relatively small. On the other hand, Table 4 indicates that the spread in CL and PMCP
among all the programs is 10 to 20 percent and 3 to 4 percent of C1, respectively. This is
due to the fact that these meanlines have relatively large slope changes along the entire
chord. The a = 0.2, a = 0.5, and a = 0.8 meanlines have a sign change in the derivative of
the slope near the trailing edge while the a = 1.0 meanline has a slope distribution which is antisymmetric about midchord. For these meanlines the spread among the results of
Table 4 suggests that it may be important to have a relatively large number of control
points located along the entire chord.
The designated meanlines of Table 5 are all characterized by having the largest negative
slope at 2 1/2 percent of chord aft of the leading edge and then a monotonically increasing
the spread among the values of CL and PMCP do not show as wide a variation as for the NACA meanlines. The spread in CL and PMCP among all six sets of results typically ranges
from 4 to 11 percent and from 5 to 8 percent of Cr respectively. Without the
Margason-Lamar results, the spread in CL and PMCP is typically 3 to 8 percent and 1 to 3 percent of Cr respectively.
It is of interest to note that for most camber cases, the use of only four chordwise con-trol points with the leading concon-trol point 11.7 percent of chord from the leading edge (the
Bandler program) gives results for CL and PMCP which are respectively within 5 percent and
2 percent of C of the results obtained by using 20 chordwise control points (the Lopez-Shen
program). The camberlines for which there is least agreement between these two programs
are the NACA 210 and 230 meanlines (which are characterized by having all the slope
changes confined to a small region near the leading edge) and the NACA a = 0.8 and
a = 1.0 meanlines (which are characterized by relatively large slope changes along the entire chord). The results obtained by using four chordwise control points but with the leading
control point 18.5 percent of chord from the leading edge (the Margason-Lamar program)
are generally substantially poorer than the corresponding results when the leading control
point is 11.7 percent of chord from the leading edge.
It is also of interest to compare the overall coefficients between the two sets of Lopez-Shen results when Il and 20 chordwise control points are used. These results show that
11 chordwise points are usually sufficient to give CL and PMCP accurate to two places. The
values of BMCP between the two sets of results agree to nearly four places. This supports
the previous statement that changes in the number of chordwise control points have
relatively little effect on BMCP.
COMPUTER RUNNtNG TIMES
Except for the Lopez-Shen program (run on an IBM 370/165) the computer times mentioned here refer to running times on the CDC 6700 computer currently in use at NSRDC. They do not include the times required for program compilation. The relative
speeds of the CDC 6700 and the IBM 370/165 are discussed in a footnote to Table 7.
Table 7 shows the range of execution times required by each program to calculate one of the planforrn cases shown in Table 3 and the time required to calculate nine camber cases such as those shown in Tables 4 and 5. The manner in which each program was used to generate results for the planform and camber cases was described earlier and summarized in
Table 1. Differences in usage of each program, such as a different number of functions or
Table 7 shows that the Wagner program has the widest variation of computer times for the planform cases. The running time depends on the planform shape, with rectangular
planforms requiring the least time. It has already been mentioned that N,M and bb/2c were
varied in several runs made with the Wagnerprogram. These runs revealed that the execution
time increases roughly linearly with the product (N)x(M) for a given value of ¿5b/2c. For
given values of N and M, execution time decreases with increasing values of &b/2c.
For camber cases, Table 7 shows the ratio of the time required to calculate nine
camber cases to the time required to calculate one planform case. For most programs, the camber cases were run with a number of other cases, and thus the times required to run
them could not be determined exactly. Table 7 indicates that there is a wide variation in
this ratio. At one extreme are the Margason-Lamar, Lamar, andTulinius programs, for which
the ratio is about equal to 9. This is expected in the case of the Margason-Lamar and
Lamar programs since they can handle only one camber case at a time. The high value of
t for the Tulinius program was quite surprising at first in view of the program option to
save the downwash matrix and not recalculate it for subsequent camber cases for a given
planform. This option was exercised for the cambercases shown in Tables 4 and 5.
How-ever, the trend observed in this study is consistent with the Langan-Wang8 study whereit was
found that to calculate one planform case and two camber cases by the Tulinius program
required more than triple the time required to calculate one planform case alone.*
At the other extreme are the Lopez-Shen and Bandler programs for which the ratio t
is only slightly greater than 1, i.e., these programs require only slightly longer to calculate
nine camber cases than to calculate one planform case. The ratio t, for the Wagner
pro-gram is 2 to 2.5. The explanation is that this program can deal with a maximum of five
camber cases at one time. Thus, two runs are required to calculate nine camber cases.
*
In a private communication, Mr. Jan R. Tulinius has indicated that computer time requirements for his program are given by:
solution time = K1 x (number of control points) x (number of vortices) +
K, x (number of spanwise functions x number of chordwise functions),3 where K1 and K2 are constants. Saving the downwash matrix for a given camber case means that the term involving K2 is Set equal to zero. Apparently, for the manner in which this program was used in the present study, the term involving K1 is dominant.
COMPARISON WITH EXPERIMENTAL RESULTS
The program results are now compared with experimental results for planform cases,
camber cases, and flap cases.16 24 The planforrn cases are compared for overall coefficients
as weil as spanwise and chordwise distributions of the loading. The comparison for the
camber and flap cases is restricted to overall coefficients. The experimental conditions for
each experiment are summarized in Table 8. Figures 3 to 6 illustrate special features of the
experimental wing configurations.
PLAN FORM CASES
Eight different planform cases are considered. Taken together, these planforms range
in aspect ratio from 2 to 8, in quarter-chord sweep from O to 56.3 degrees, and in taper
ratio from O to 1. In addition, wing-body combinations are considered as well as wing
alone.
Five of the planforrns are part of the 60systematic planforms discussed previously. The
computer results for these five planforms are shown in Tables 3.3, 3.18, 3.33, 3.47, and
3.48. Table 9 gives the experimental overall coefficients corresponding to these cases.
Tables 10, 11, and 12 show the computer and experimental overall coefficients corresponding
to three other cases.
It may be noted that the computer results for the five experimental cases shown in
Table 9 were obtained for Mach number Ma 0. In addition, the Lopez-Shen and Bandler
18Wjck B.H., "Chordwise and Spanwise Loadings Measured atLow Speed on a Triangular Wing Having an Aspect Ratio
of Two and an NACA 0012 Airfoil Section," Ames Aeronautical Laboratory,NACA Technical Note 1650 (Jun
1948).
19Smith. D.W. and V.D. Reed, "A Comparison of the Chordwise Pressure Distribution and Spanwise Distribution of
Loading at Subsonic Speeds of Two Triangular Wings of Aspect Ratio 2 Having NACA 0005 and 0008 Sections,"
Ames Aeronautical Laboratory, NACA RM A51L21 (May 1952).
20Falkner, V.M. and D.E. Lehrian, "Low-Speed Measurementsof the Pressure Distribution at the Surface of a
Swept-back Wing," Aeronautical Research Council (Great Britain) R. & M. 2741 (1953).
21Jacobs, W., "PressureDistribution Measurements on Unyawed Swept-Back Wings," NACA TM 1164 (Jul 1947).
22Jacobs, W., "Systematische Druckverteilungsmessungen anPfeliflugeln konstanter Tiefe bei symmetrischer und
unsymmetrischer Anstromung," Il. Mitteilung, Ingenieur Archiv, Vol. 19, pp. 83-102 (1951).
23Graham, R.R., "Low-Speed Characteristics of a45° Sweptback Wing of Aspect Ratio 8 from Pressure Distributions
and Force Tests at Reynolds Numbers from 1,500,000 to 4,800,000," Langley Aeronautical Laboratory, NACA RML51H13
(1951).
24Boyd J.W., "Aerodynamic Characteristics of Two 25-Percent-Area Trailing-Edge Flaps on an Aspect Ratio 2
programs, which cannot handle Ma corrections, were also run at Ma = O for the cases shown in Tables 1 0, II, and I 2. The experiments were, of course, conducted at nonzero Ma, with
the maximum value (Ma 0.24) occurring for the Smith and Reed19 experiment shown in
Table 11. In order to determine the effect of Mach number, the Tulinius program was run
for both Ma = O and the experimental values for the cases shown in Tables 10 through 12. Table 10 indicates that even for Ma = 0.24, the two sets of values for the Tulinius program differ by less than 1 percent for CL, 0. II percent of Cr for PMCP, and 0.04 eprcent of semi-span for BMCP. Lower values of Ma lead to correspondingly smaller differences. Thus, the Mach number correction is not significant for the planform cases considered here.
Overall Coefficients
Comparison of the overall coefficients shows that except for the Bandler program and-to a lesser extentthe Lopez-Shen program in the case of CL, the computer results generally agree well with the experimental results. The values of CL, PMCP, and BMCP are usually predicted to within 3 to 4 percent, 1 percent of C1, and 2 percent of b/2, respectively. The Lopez-Shen program predicts values of CL which are usually higher by O to 14 percent than the experimental values. Tables 10 and 12 show that a 3 to 4 percent difference in CL is less than the variation in CL due to a change in Reynolds number or in the airfoil section used. The variation due to the latter two changes may be as high as 1 0 percent. Tables 9 to
12 also indicate that the values of CL from force and pressure measurements typically dis-agree by about 5 percent. The Bandler program predicts values of CL and PMCP which typically differ from the experimental values by 5 to 6 percent and 3 to 5 percent of C1, respectively. It also predicts values of BMCP which are usually several tenths of a percent of b/2 more in error than any other program.
It is interesting to compare in some detail the computer results for swept untapered wings of aspect ratio 5 (Tables 3.18, 3.33, and 3.48) with the corresponding results shown in Table 9. The presence of the P-functions used at the wing root, where there is a break in the wing planform, causes the Tulinius program to yield the lowest computed values of CL. For the 1 5- and 30-degree sweeps, the Tulinius program is in closer agreement with the ex-perimental values than any of the other programs; the reverse is true for the 45-degree sweep case. This suggests that the P-functions improve the accuracy of the program for moderate sweeps of less than 35 or 40 degrees, but decrease the accuracy at higher sweeps.
The one notable exception to the good agreement in CL between computer and
experi-mental results is the Wick18 experiment shown in Table 10. Here, the computer results for
