Post on 21-Jan-2023
AN APPROACH TO THE DESIGN
OF AIRFOILS WITH HIGH LIFT TO DRAG RATlOS
NOVEMBER, 1983
BY
DAVID WALTER ZINGG
TECHNISCHE HOGESCHOOL D[lFT -LUCHTVAART- EN RUIMTEVAARTTECHNIEK
BIBLIOTHEEK Kluyverweg 1 - DELFT
UTIAS TECHNICAL NOTE NO. 245 CN ISSN 0082-5263
•
t
AN APPROACH TO THE DESIGN
OF AIRFOILS WITH HIGH LIFT TO DRAG RATlOS
BY
DAVID WALTER ZINGG
SUBMITTED MARCH~ 1983
NOVEMBER, 1983 UTIAS TECHNICAL NOTE NO. 245 CN ISSN 0082-5263
Summary
A procedure for the design of low-speed, single-element airfoi1s with high lift to drag ratios is presented. The p r ocedure uses an inverse approach which proceeds from a set of desiraó1e boundary 1ayer characteristics, which are determined from the performance objectives, to a velocity distribution and, fina11y, to an airfoi1 shape. The boundary 1ayer shape factor, whi ch is a measure of the nearness of the boundary 1ayer to separation, is used to de fine the boundary 1ayer characteristics in the upper surface pressure recovery region. The turbulent boundary 1ayer equations are solved in inverted form to ca1cu1ate the velocity distribution which corresponds to a shape factor variation. The ve10cities on the upper surface prior to pressure recovery are chosen to cause boundary 1ayer transition at a desired location. The lower surface velocity distribution is se1ected to satisfy the requirement that a c1osed, nonreentrant shapewhich meets the specified structura1 requirements resu1ts. The design procedure described can be uti1ized for the design of high performance airfoi1s for a variety of performance objectives, operating conditions, and practical constraints.
Using this procedure, an airfoi1 has been designed which achieves a lift to drag ratio of 180 at a lift coefficient of 1.2 and a Reyno1ds number of one mi11ion. Through this design study, a set of guide1ines for the design of a1rfoi1s with high lift to drag ratios is estab11shed.
ii
',..
AcknowTedg'ement s
The author wishes to thank Dr. G. W. Johnston of
the University of Toronto Institute for Aerospace Studies
for his supervision. In addition, the author is grateful
to Mr. B. Egglestone of de Havilland Aircraft for his
advice and for the use of his computer programs.
iii
•
't
Ta'bleof Contents
Page
Summary ii
Acknow1edgements iii
List of Figures v
List of Symbols vi
1. Introduction 1
2. Basic Design Considerations 4
3. Design Procedure 6
4. Resu1ts and Discussion 23
5. Conclusions 35
6. References 39
Figures 42
Appendix A: Thwaites' Method for the Laminar Boundary Layer Calculation 53
Appendix B: Inverse Turbulent Boundary Layer Program 55
Appendix C: Program to Calculate the Gradient of a Linear Velocity Distribution to Give a Desired Transition Point 57
Appendix D: Coordinates of Designed Airfoil 58
iv
. L~&~bf F~gures
Fig. 1. Drag on a flat p1ate with different transition points
Fig. 2. Velocity gradients fo r transition at x/c = 0.7
Fig. 3. Shape factor variation on the NAGA 633-018
Fig. 4. Boundary 1ayer characteristics of Liebeck L1003 and Wortmann FX 74-GL6-140 airfoi1s
Fig. 5. Experimenta1 lift, drag, and pitching moment data for Liebeck L1003 airfoi1
Fig. 6. Deve10pment of Rg and Rgtr for a constant velocity
distribution with Re = 2.0 X 10 6
Fig. 7. Fami1y of upper surface velocity distributions
Fig. 8. Variation of lift coefficient and lift to drag ratio with recovery point 10cation
Fig. 9. NAGA 67-015 thickness form cambered to give design upper surface velocity distribution with xr/c = 0.7
Fig. 10. Velocity distribution on above section at ~ = 30
Fig. 11. Designed airfoi1 section
Fig. 12. Velocity distribution on designed section at design incidence, ,~= 3.50
Fig. 13. Shape factor variation on designed section at design incidence
Fig. 14. Momentum thickness on designed section at design incidence
Fig. 15. Velocity distribution on designed section at ~= 1.00
Fig. 16. Ga1cu1ated lift, drag, and pitching moment 6 characteristics of the designed airfoi1 at Re = 10
Fig. 17. Velocity distribution on designed section at ~ = 40
Fig. 18. Velocity distribution on Liebeck L1003 airfoi1 at design incidence
v
, .
'.
c
H
1
LID
p
Re
u
UQ()
x r
f
List Of Symbols
chord length
drag coefficient
local skin friction coefficient
lift coefficient
pitching moment coefficient about quarter-chord
pressure coefficient
.boundary layer shape factor, ~ ;9
parameter in Thwaites' method
lift to drag ratio
pressure
chord Reynolds number
Reynolds number based on momentum thickness
Reynolds number based on streamwise distance
airfoil thickness
streamwise velocity at outer edge of boundary layer
free-stream velocity
position of recovery point
angle-of-attack
boundary layer displacement thickness
parameter in Thwaites' method
kinematic viscosity
density
vi
SUb scripts
i
te
tr
ooundary layer momentum thickness
wall shear stress
denotes a value at the start of the recovery region
denotes a value at the trailing edge
denotes a value at the transition point or a value required for transition
vii
.'
1. Introduction
The advent of high-speed computers and advanced
numerical methods has led to new possibi1ities in airfoi1
design. Prior to 1930, airfoi1 design was primari1y an
empirica1 process aided somewhat by potentia1 flow theory.
In the 1~30's, boundary 1ayer considerations were
qua1itative1y integrated into the design of "low-drag"
sections such as the NACA 6-series airfoi1s. Significant
improvements upon these sections were not achieved unti1
the late 1950's, when large computers became avai1ab1e.
Since that time, the performance of the NACA 6-series has
been consistent1y exceeded by sections such as the Wortmann
FX-series (Ref. 1).
The present~day designer has at hand an extensive array
of numerical ca1cu1ation techniques. Singu1arity methods
are a general, accurate, and efficient way of performing
a potentia1 flow ca1cu1ation. Boundary 1ayer characteristics
can be accurate1y determined by a variety of techniques,
inc1uding integra1 and finite-difference methods (which
invo1ve some empiricism). Empirica1 corre1ations adequately
predict boundary 1ayer transition and separation. Fina1ly,
iterative schemes are avai1ab1e to combine potentia1 flow
and viscous solutions so the ana1ysis of ful1y attached
1
viscous flow about two-dimensional sections can be
performed.
The recent advances in airfoil design, however, are
primarily the result of a change in the fundamental
design approach from an iterative cut and try procedure
to a synthesis or inverse technique. The traditional
empirical approach consists of modifying an initial
shape by trial and error until given performance
objectives are achieved. The inverse approach, in
contrast, begins with a set of boundary layer characteristics
which are derived from the performance objectives. The
velocity distribution which results in these characteristics
is determined and, finally, the shape which produces this
velocity distribution is calculated. The airfoil section
shapes generated by such an inverse design approach, such
as the Liebeck high-lift sections (Ref. 2), are generally
unique and are not likely to be obtained using an
empirical approach.
This thesis describes an example of an inverse approach
applied to the design of an airfoil with a high lift to
drag ratio. As energy costs continue to rise, low fuel
consumption is becoming an increasingly high priority
in aircraft design and hence such sections are required.
In general, an airfoil design problem involves several
conflicting requirements and therefore the final solution
2
must include some compromises. The design criteria and
their relative importance must be determined from the
intended application. While the airfoil designed in this
study reflects the specific priorities chosen, the design
procedure described can be applied to the design of
airfoils for a variety of applications. Some basic airfoil
design considerations are discussed in the following
section.
3
2. Basic Design Considerations
The design of a new low-speed, single-element
airfoil involves finding a section with desired performance
characteristics under specific operating conditions within
certain practical constraints, i.e. a section tailored to
a given application. However, because there are many
design trade-offs, different section shapes can be found
for a single application, each the re sult of different
priorities.
The performance objectives normally considered are
high lift, high lift to drag ratio, low drag, or a
specific pitching moment coefficient. When high lift is
desired, such as for STOL aircraft, the importance of
the nature of the stall break must be assessed, i.e.
should the maximum lift coefficient be sacrificed somewhat
to achieve "gentle" stall? Similarly, if a high lift to
drag ratio is the goal (or simply low drag, such as for
a strut), a compromise must be reached between the minimum
drag and the width of the low drag bucket, since a
decrease in drag is generally accompanied by a reduction
in bucket width. Possible pitching moment characteristics
which may be desired, though usually a low priority, are
a positive moment for flying wing type aircraft or a low
4
, .
------------ - - -- - ----------- - -------- ------ ------
moment to reduce trim loads and aeroelastic effects.
The operating conditions which must be specified are
the design lift coefficient, Re ynolds number, and Mach
number. In addition, off-des ign con siderations lead to
a range of these quantities under which adequate performance
is expected.
Practical constaints include a structural requirement
such as a minimum section modulus. Of ten simply a minimum
thickness/chord ratio is stipulated. Also, the type of
wing construction is a practical consideration for the
surface quality has a significant effect on the location
of boundary layer trans ition.
The criteria for this design study were not chosen
with a specific application in mind but rather to prove
the validity and accuracy of the design procedure in
general. The design goal was to maximize the section lift
to drag ratio with no restriction imposed upon the design
lift coefficient. Minimal consideration was given to
off-design performance and the stall break. The design
Reynolds number was chosen to be one million corresponding
to the aval1able test facillty. Compresslbility effects
were neglected. The thickness/chord ratio was required
to be fifteen percent, typlcal of a practical section,
and the surface was assumed to be smooth.
5
3. Design Procedure
An inverse airfoil design approach begins with a set
of boundary layer characteristics which are determined
from the performance objectives. The boundary layer
characteristics of primary interest are the location of
transition and the local proximity to separation in the
pressure recovery region, i.e. the angle-of-attack
perturbation which can be tolerated without flow sepa
ration. Calculating the velocity distribution which
results in the desired boundary layer characteristics
constitutes the inverse boundary layer problem. The
d"irect problem is to solve the boundary layer equations
for the boundary layer parameters, H, Q, and Cf' given
a velocity distribution. In inverted form, these equations
can be solved for a velocity distribution if the variation
of one of the parameters is specified.
The airfoil shape is determined from the calculated
velocity distribution by an inverse potential flow
calculation. However, a velocity distribution can be
produced by a real, i.e. closed and non-reentrant, shape
only if certain conditions derived from conformal mapping
theory are satisfied (Ref. 3). The structural requirements
introduce a further constraint. Therefore the boundary
6
layer characteristics cannot be specified on the entire
airfoi1. Hence it is practical to define a form for the
velocity distribution, exp1icit1y uti1izing the inverse
boundary 1ayer equations on1y in regions where the boundary
1ayer characteristics are particu1arly important.
For most app1ications, the upper, or 1ifting, surface
is more critical. Thus the upper surface velocity distri
bution can be defined whi1e the lower surface ve10cities
are allowed some freedom in order that a physica11y
rea1izab1e shape which satisfies the structura1 requirements
wil1 result. On the upper surface, the boundary 1ayer
deve10pment in the pressure recovery region is extreme1y
important. In this region the boundary 1ayer characteristics
shou1d be specified and the ve10cities exp1icit1y ca1cu1ated.
Prior to pressure recovery, a form is norma11y assumed
for the velocity distribution, though the inverse laminar
boundary 1ayer equations can be app1ied in the transition
region to achieve the desired location and type of
transition.The construction of the upper surface velocity
distribution, inc1uding the specification of the transition
point and the boundary 1ayer characteristics in the
recovery region, represents the key element in the design
process. The remainder of the prob1em is to a large extent
computationa1.
The performance objective in this study is a high lift
7
to drag ratio. The primary consideration in the design
of a low drag airfoil is the location of the transition
from laminar to turbulent flow. As shown in Figure 1, the
skin friction drag of a turbulent boundary layer is much
greater than that of a laminar layer. Thus transition
must be delayed as long as possible on both surfaces of
the airfoil.
The transition point location is dependent primarily
upon the Reynolds number and the pressure distribution.
Secondary influences are the free-stream turbulence level
and the roughness of the airfoil surface. In general, an
increase in Reynolds nurnber reduces the stability of the
laminar boundary layer and a negative velocity gradient
adds a further destabilizing influence. These effects are
shown in Figure 2, which displays three linearly varying
velocity distributions which re sult in the same transition
point (at x/c=0.7, where u/u~=1.5) at three different chord
Reynolds numbers. At a Reynolds number of 1.5 X 10 6 , a
constant velocity leads to transition at x/c=0.7. In order
to achieve the same transition point at a Reynolds nurnber
ten times higher, the stabilizing effect of a positive
velocity gradient is required while a negative gradient
canbe tolerated at a Reynolds number ten times lower.
For the usual range of chord Reynolds numbers, the boundary
layer cannot remain laminar in a region of substantial
8
adverse pressure gradient. Thus a turbulent boundary
layer is inevitable on the aft portion of the upper
surface of a lifting airfoil without active boundary
layer control as the pressure recovers to its trailing
edge value.
The rate at which the pressure can recover is limited
by the possibility of turbulent separation which results
if the adverse pressure gradient is too steep and must
be avoided on a low drag airfoil. Therefore the shortest
possible distance for a given velocity decrease is that
for a velocity distribution which is on the verge of
separation throughout. Such velocity distributions, which
have a concave shape in contrast to the roughly linear
recovery regions of the NACA 6-series, have been employed
in the design of high lift sections (Ref. 2). If transition
is avoided until pressure recovery begins, the length of
the turbulent boundary layer region is minimized by this
form of pressure recovery. Alternatively, for a given
transition point and hence a given length for the pressure
recovery region, such a pressure distribution allows the
highest possible velocity prior to recovery to a given
trailing edge velocity. Hence this form of pressure recovery
is suitable for the design of airfoils with high lift to
drag ratios as weIl. From a practical standpoint, though,
the flow about an airfoil should not be on the verge of
9
separation when the airfoil is operating at its design
incidence. However, a similar argument also applies to
a velocity distribution which has a cons tant local
proximity to separation in the recovery region. For a
given danger of separation, such a pressure recovery
region, which is also concave, results in the shortest
possible recovery region and hence a high lift to drag
ratio.
A pressure distribution with a constant local proximity
to separation in the recovery region leads to a sudden
stall break. In general, it is preferabIe for the onset
of separation .to begin at the trailing edge and to proceed
forwards on the airfoil as the angle-of-attack is further
increased. Therefore, at the design incidence, the boundary
layer should become nearer to separating as the trailing
edge is approached.
Several criteria exist for predicting turbulent
separation (Ref. 4), including Stratford's method (Ref. 5)
and a shape factor criterion which predicts separation
when H reaches a value between 1.8 and 2.4. A pressure
distribution which exactly satisfies Stratford's criterion
throughout can be calculated without recourse to the
inverse boundary layer equations, as given by the following
equations (the "Stratford distribution", Refs. 5 and 6):
10
,-'
1
C "T 1-a/(x+b)~, p
- n-2 C ~-P - n+l (3.1 )
(3.2)
n =
s = o
10glORo
38.2(VIXtrUtr)3/8(Uo/Utr)1/8[~Xtr(U/Uo)5d(it~5/8Xtr o r
+ ( x 0 ( ui u ) 3 dx Jx 0 tr
The constants a and bare determined by the requirement
of continuity between equations 3.1 and 3.2. The subscript
o denotes a quantity at the start of pressure recovery. The
quantity So accounts rough1y for a favourable pressure
gradient or a laminar boundary layer prior to pressure
recovery.
The shape factor criterion requires a complete boundary
layer calculation and hence takes into account more
accurately the boundary layer development prior to the
recovery region (Ref. 4). A velocity distribution can
be calculated from a shape factor variation through the
inverse turbulent boundary layer equations. Since the
value of the shape factor quantifies the local proximity
to separation, its use allows considerably more flexibility
11
than the use of Stratford's criterion. The shape factor
thus provides an ideal link between the desired boundary
layer characteristics and the velocity distribution in
the recovery region. Figure 3 displays the shape factor
variation of a NACA 6-series section, which is typical
of roughly linear pressure recovery. The boundary layer
becomes much nearer to separating as the trailing edge
is approached, particularly at the lowest Reynolds number
shown. Steeper recovery is possible early in the recovery
region without a reduction in the angle-of-attack at which
the airfoil stalls.
The criterion that separation occurs when H reaches a
value between 1.8 and 2.4 was developed for conventional
shapes, which generally have a rapidly increasing shape
factor in the vicinity of the separation point. For a
roughly constant shape factor, the critical value appears
to be close to 2.4. The shape factor development of a
Liebeck high-lift section at its design incidence (Fig. 4(a)),
which is typical of aStratford velocity distribution, has
a constant value of H of roughly 2.2 in the recovery
region. Yet the stall for this section, which is extremely
sudden, occurs at an angle-of-attack about four degrees
above the design incidence (Fig. 5). The shape factor
development of an airfoil designed for a more gradual
stall break typically has higher values of H in the
12
vicinity' of the trailing edge, as shown in Figure 4 (b) .
The use of the shape factor as the input to the inverse
Doundary' layer equations is ideal for the design of an
airfoil with given stall characteristics.
Since the nature of the stall break was not a high
priority in this particular design study, a constant
value of H equal to 1.8 was selected for the pressure
recovery region. While a higher value would lead to
steeper recovery and hence improved performance at the
design incidence, this value was considered to be more
suitable for a practical section, of which adequate
performance at off-design Reynolds nurnbers and angles-of
attack is expected.
A linear variation in velocity was chosen to precede
the pressure recovery region, the slope to be determined
such that boundary layer transition occurs in an orderly
fashion just prior to the start of recovery. Transition
by laminar separation and reattachment, although used
successfully by Wortmann (Ref. 7), can lead to an initially
thick turbulent boundary layer. Orderly transition can
be predicted with reasonable accuracy by the simple
correlation given by Cebeci et al (Ref. 8) which accounts
for the effects of the pressure distribution and the
Reynolds nurnber only. Transition is predicted when the
Reynolds number based on the momentum thickness of the
13
laminar boundary layer exceeds the value calculated from
the Reynolds number based on the steamwise distance by
the following formula:
Figure 6 shows the development of RQ and RQtr for a
constant velocity distribution at a chord Reynolds number
6 of 2 X 10 , transition being predicted at x/c = 0.8.
Using the approach described in Reference 9, an inverse
turbulent boundary layer program was written to determine
a velocity distribution from a specified shape factor
variation (Appendix B). If the shape factor variation is
given, three equations are required to solve for the
unknowns u, Q, and ~ . Garner's equation for the shape o
factor is (Ref. 10):
This equation can be manipulated to give:
O.0135(H-1.4)u R 1/6Q
Q
(3.4)
(3.5)
The momentum integral equation for the boundary layer can
be written as:
14
----.~--~--------~------~~--------------------------------------------~
1:0 .. CH+.2 }Q~ ou2 - u
(3.6) dQ -= dx
;
The wall shear stress coefficient is determined from the
following empirical algebraic expression~ the Ludwig-
Tillman equation, which completes the set of equations:
Lo _ b.123R~O.268 X lO-O.678H -2 - '='
·~u j
Using equation 3.7~ equations 3.5 and 3.6 are integrated
numerically by a Runge-Kutta methode
Besides the H variation, additional input data required
by the program are the location of the start of the recovery
region~ the chord Reynolds number~ and the trailing edge
velocity. Initial values of the velocity and the momentum
thickness, needed to begin the numeri cal integration~ are
computed by the program. Some iteration is required to
determine the initial velocity which results in the desired
t~aillng edge velocity. Since it is assumed that transition
occurs just prior to the recovery region~ the initial value
of the momentum thickness can be determined from the
transition criterion once the velocity at transition is
known. The transition Reynolds number based on the momentum
thickness is calculated from equation 3.3; the momentum
thickness is then given by:
15
(3.8)
For the more general case in whlch the bbundary layer
becomes turbulent before the start of the recovery region,
the boundary layer development must be calculated up to
the start of recovery and the resulting momentum thickness
must be input to the program.
A second computer program was written to determine
the gradient of the linear portion of the velocity
distribution which results in the desired transition
point, given the velocity at transition found from the
p~ogram described above (Appendix C). Thwaites' method
is used to calculate the laminar boundary layer development
(Ref. 11, Appendix A). This program is also iterative,
adjusting the gradient until the necessary momentum thick-
ness for transition is achieved at the desired point. The
chord Reynolds number, the transition point, and the
velocity at transition comprise the input data. The program
is not restricted to linear velocity distributions, since
Thwaites' method is applicable to laminar flows in general.
Having selected a general form for the velocity distri-
bution and a shape factor variation in the recovery region,
two variables remain to be specified to allow' the calculation
of the upper surface velocity distribution, the trailing
16
edge velocity and the recovery point. A1though the
trailing edge velocity is severe1y constrained, it can
have a powerful effect on the performance of a section
(Ref. 12). However, the effects of varying this parameter
were not examined in this study. A trailing edge velocity
of 0.95, which is typica1 of existing sections, was input
to the inverse turbulent boundary layer calculation.
In order to investigate the effect of the recovery
point location, a family of upper surface velocity
distributions was calculated with the recovery point
varying from x/c= 0.30 to x/c= 0.90 (Fig. 7).
On the lower surface, two basic features are generally
desirabIe for the velocity distribution. The velocity
should be minimized to contribute to the section lift
coefficient and to keep the boundary layer thin, thereby
reducing the section drag coefficient as weIl. Unlike
the upper surface, there is no trade-off between lift
and drag on the lower surface, except possibly with
regard to the transition point. In addition, the velocity
should increase as long as possible to delay transition.
The lower surface velocity distribution can also be used
to achieve desired pitching moment characteristics.
However, since the overall velocity distribution must
be achieved by a real shape, the lower surface velocity
distr1bution cannot be specified independently of that
17
on the upper surface. ~or example, on a fifteen percent
thick airfoil, the mea,n of' the upper and lower surface
velocity ratios, u!u_, must in gener al be greater than
1.1 from roughly x/c=O.lO to x/c=O.60. Referring to
figure 7, the design upper surface velocity- ratio in
this region is about 1.2 ~or a recovery point at x/c=O.90.
Thus the lower surface velocity cannot be less than the
freestream valuein this region so the lift coefficient
which can be achieved by a section with this recovery
point location is limited. Therefore different upper
surface velocity distributions can only be properly
compared if their effect on the lower surface velocities
is accounted for.
An initial lower surface velocity distribution which
results in a real shape can be determined by recambering
a thickness form with the desired maximum thickness/chord
ratio to achieve the given upper surface velocity distri
bution. While the resulting lower surface boundary layer
characteristics are generally unsatisfactory, this lower
surface velocity distribution provides a basis from which
an improved distribution can be found.
In order to investigate the effect of the location of
the recovery point on the section lift coefficient and
lift to drag ratio, NACA thickness forms with (t/c)max=O.15
wereeambered to give the upper surface velocity
18
..
's
dlstributlons s-h.own in figure 7. Whenever possible, a
thicknes-s' t'orin was us-ed which begins press-ure recovery
at the same point as the design velocity distribution.
Therefore the NACA 63-015 th1ckness- form was recambered
to give the design velocity distribution wlth recovery
point at x/c=0.3, the NACA 64-015 form was used for the
distribution with recovery point at x/c=0.4, etc. For
recovery points aft of x/c=0.7, the NACA 67-015 thickness
form was used. The computer programs described in Refs. 13
and 14 were utilized for the analysis and design computa
tions. To allow for an acceleration region, the velocity
was not specified on the first three percent of th.e chord.
The design program assumes inviscid flow; hence the designed
sections do not achieve the design upper surface velocities
exactly when viscous floweffects are included in the
analysis. Also, due to a shortcoming of the design program,
upper surface transition occurs somewhat prior to the
start of pressure recovery on all of the sections, causing
higher drag values than expected. Figure 8 shows the lift
coefficients and lift to drag ratios of the designed
sections as a function of recovery point location.
If a design lift coefficient were specified, the recovery
point location would be chosen to satisfy this requirement.
Note that this design approach is not applicable to an
extremely low design lift coefficient since th1s would
19
require that the pres,sure recovery reglon on the lower
surf'ace als-o De cons-ldered in detail. As the lift
coefficient was not constrained in this study, the
recovery point location which results in the maximum
lift to drag ratio could De chosen. Hence the section
with recovery point at x/c=0.7 was selected for further
study and refinement.
Figures 9 and 10 display this section and the correspond
ing velocity distribution. The upper surface trailing edge
velocity ratio is considerably lower than the design
value of 0.95. The trailing edge velocity is constrained
by limitations on the physical shape of the trailing edge
region and by viscous effects. In order to achieve the
exact H variation specified, the trailing edge velocity
obtained should be input into the inverse boundary layer
program, resulting in a new velocity distribution, a new
section shape, and a new trailing edge velocity. This
iterative procedure usually converges satisfactorily
af ter one or two iterations if the initially specified
trailing edge velocity is reasonably weIl chosen. In
this case, however, the effect of the low trailing edge
velocity is to cause an increase in the shape factor
near the upper surface trailing edge. Since this should
improve the section stall break, no iteration was performed.
The lower surface velocity distribution has the desired
20
~'
characterist;tcs: unt1.1 roughly- x!c=.O.7 CF;tg. io). Since
the upper surface velocity initia11y recovers more quick1y
than on the NACA 67-015 and the mean velocity is constrained
by the thickness form,there is a corresponding increase
in the lower surface velocity af ter x!c=0.7. This loca1
velocity increase reduces the section lift coefficient
and causes premature boundary 1ayer transition, thereby
increasing the drag coefficient. Referring to figure 9,
the physica1 cause of this velocity increase is apparent:
there is a sma11 protruberance on the lower surface of
the airfoi1 between x/c=0.7 and x/c=0.9. Remova1 of this
protruberance and hence the associated velocity perturbation
shifts the transition point to x/c=0.89 from x/c=0.76,
improving the section lift to drag ratio by about seven
percent. The a1tered thickness form must be slight1y
recambered to restore the upper surface velocity distribution.
At ang1es-of-attack above the design incidence for
this section, a velocity peak resu1ts on the upper surface
near the 1eading edge, causing transition by 1aminar
separation and reattachment. Since a turbulent boundary
1ayer grows more quick1y than a 1aminar one, this ear1y
transition resu1ts in a thicker boundary 1ayer at the
recovery point. The thicker boundary cannot to1erate the
steep gradient at the beginning of the pressure recovery
region 50 the flow separates. In order to improve the
21
off-design performance, the airfoi1 was redesigned to
give a flat velocity distrioution near the 1eading edge
at an ang1e-of-attack rough1y three degrees above the
design va1ue. Thus at the design incidence, the ve10cities
near the 1eading edge are reduced, the flow acce1erating
more gradua11y to its peak ve loci ty (compare Figs. 10
and 12). This reduces the lift to drag ratio somewhat.
The lower surface was simi1ar1y redesigned near the
1eading edge to remove the velocity peak 2.5 degrees
be10w the design incidence (Fig. 15).
The a1teration of the upper surface ve10cities near
the 1eading edge affects the location of the transition
point. A1so inaccuracies in the design program resu1t in
premature transition. Therefore the fina1 step in the
design process was to modify the velocity distribution
on the upper surface in the vicinity of the transition
point by trial and error unti1 transition occurs at the
desired location. This reduced the section drag by ten
percent. Figures 11 and 12 show the fin al section shape
and the velocity distribution at design incidence.
22
Ol
i.
4' • Resultsand Dis·cus.s.1on
Figure 16 displays the calculated lift, drag, and I
pitchirig moment characteristics of the designed airfoil
at the design. Reynolds number of one million. The computer
program used is applicable to the analysis of fully
attached flow only (Ref. 14). At angles-of-attack above
four degrees, turbulent separation is predicted on the
upper surface while at angles below zero degrees, laminar
separation occurs at the lower surface leading edge
without subsequent reattachment. Since the performance
in these regions could not be calculated, the dotted
lines on the figure roughly indicate the expected trends.
The skin friction drag remains roughly constant
throughout the angle-of-attack range for attached flow,
contributing roughly 2/3 of the drag at the design incidence.
The maximum section lift to drag ratio, which occurs
at a lift coefficient of 1.2, is over 180. This value is
higher than that achieved at a Reynolds number of one
million by sections designed in the earlier empirical
manner and most modern sections as weIl, such as those
designed by Wortmann for a similar performance objective
and similar operating conditions, described in Reference
15. However, the off-design performance is inferior to
23
that of many' sectlons of the same thickness, such as
those of Wortmann. Thes'e resul ts are a reflection of
the design priorities, since the performance objective,
a high lift te drag ratio, was emphasized while minimal
consideration was given to off-design performance. However,
even in light of these priorities, the airfoil's performance
is not entirely as desired.
The problem is that the airfoil begins to stal1 at an
angle-of-attack only 1.5 degrees above the design incidence.
The highe st lift to drag ratio experimentally verified
at a Reynolds number of one million (to the author's
knowledge) is about 220, achieved by the 1iebeck high-lift
airfoil, 11003, shown in figures 4(a) and 5 (Ref. 2). This
value occurs at the maximum lift coefficient, i.e. just
prior to the onset of stall. At the design incidence,
roughly four degrees lower, a lift to drag ratio of 180
is obtained. Therefore, no advance can be claimed until
either a lift to drag ratio greater than 220 is achieved
or a somewhat lower value is obtained with an increased
margin from stall. An example of the latter would be an
airfoil which has a lift to drag ratio of 180 at an
angle-of-attack more than four degrees below the value
at which stall cemmences.
As shown in Figures 4(a) and 13, the shape factor in
the recovery region of the designed airfoil is roughly
24
1.8 at the design incidence whereas on the Liebeck airfoi1
it is about 2.2. Thus it might be expected that the designed
airfoi1 wou1d have an increased margin from sta11. However,
while the Liebeck airfoil can tolerate an increase in
angle-of-attack of four degre e s, the flow separates on
the designed section on1y 1.5 degrees above the design
incidence. Clear1y this separation is not caused simply
by the pressure gradient in the recovery region. Rather
it is caused by a shift in the upper surface transition
point forwards from the recovery point, resulting in a
thicker boundary 1ayer which cannot withstand such steep
pressure recovery. The transition point moves from x/c=O.69
at the design incidence,a=3.5°, to -x/c=o.63 at~=4°, where
a consequent drag rise is seen (Fig. 16), to x/c=O.53
at ct=5°, where separation first occurs.
Figure 17 shows that an increase in the section incidence
above 3.50 resu1ts in an increased negative velocity
gradient in the laminar region of the velocity distribution.
Hence the 1aminar boundary 1ayer thickens more quickly
and is prematurely destabilized, causing transition prior
to the recovery point. At angles-of-attack below 30, the
velocity gradient is reduced so the boundary layer does
not transition prior to the recovery point. Thus a laminar
separation bubble is formed when the laminar boundary
1ayer is exposed to the steep initial pressure gradient
25
in the recovery region. Accurately calculating the
effect of this bubble, which results from laminar
separation and subsequent reattachment, is beyond the
capabilities of the computer program used so the results
shown in Figure 16 include only a rough approximation.
In order to restriet the movement of the transition
point with changes in angle-of-attack, the destabilization
of the laminar boundary layer must be accomplished over
a much shorter region. For example, at its design
incidence, the Liebeck airfoil has a positive velocity
gradient until just short of the recovery point, where
a transition ramp with a negative velocity gradient is
used to provoke transition (Fig. 18). As the angle-of
attack is increased, the positive velocity gradient is
reduced but not sufficiently to cause transition prior
to the transition ramp. At lower angles-of-attack, the
transition ramp is steep enough to ensure that transition
occurs before the recovery point, thereby avoiding a laminar
separation bubble. Thus the location of the transition
point is confined to the transition ramp for a wide range
of angles-of-attack.
At low Reynolds numbers, such as one- million, the
laminar boundary layer is difficult to destabilize. Hence
a substantial negative velocity gradient is required to
provoke transition within a sufficiently short distance.
26
A 1engthy transition ramp does not fu1fi1 its purpose,
which is to minimize the movement of the transition point
with changes in ang1e-of-attack. However, the negative
velocity gradient needed f or a short ramp cou1d cause the
1aminar boundary 1ayer to separate.
The inverse 1aminar boundary 1ayer equations must be
solved to determine the velocity distribution for the
transition ramp which resu1ts in a given shape factor
variation. Reference 9 gives the fo110wing simp1ified
solution:
u -u ,y:;
(1-H/2.55) C(~) 0.94
c (4.1)
and the subscript ( )0 denotes a va1ue at the start of
the ramp.
Laminar separation is predicted when the shape factor
reaches a va1ue of 3.5. Thus the steepest possible ramp
without separation has a constant shape factor with a
value slightly under 3.5, analogous to a pressure recovery
region which is on the verge of separation throughout.
Substituting a shape factor of 3.5 into equation 4.1 gives:
u = u oo
(U/U~)o (x/c)-0.40 (x/c).-0.40
o
27
(4.2)
A lower value of H must be chosen at the design
incidence so that laminar separation is avoided over a
range of angles-of-attack. A more practical value such
as 3.2 results in:
u uD<; =
(x/c)-O.27 o
(x/c)-O.27 (4.3)
From equation 4.1, the negative velocity gradient for
a given shape factor greater than 2.55 is reduced for a
tiansition ramp which is located towards the aft end of
the airfoil section. Since it is difficult to destabilize
a laminar boundary 1ayer at low Reynolds numbers, the
combination of a low Reynolds number and a late recovery
point therefore causes an excessively long ramp to be
required to provo~e transition if laminar separation is
to be avoided.
Under these circumstances it may be desirabIe to fix
the transition point through the use of a controlled
laminar separation bubble as in Figure 4(b). The transition
ramp required required to cause a small bubble can be
determined by substituting a value of H slightly above
3.5 into equation 4.1. The exact value of H at the design
point again depends on the range of angles-of-attack at
which transition must be provoked. While this approach
leads to minimal movement of the transition point with
28
changes in angle-of-attack, the laminar separation bubble
can be detrimental to performance if it becomes too large.
Also the bubble cannot be allowed to burst at high angles-
of-attack before turbulent separation has begun in the
recovery region.
By restricting the movement of the upper surface
transition point, the performance of the designed airfoil
at angles-of-attack above the design value could be
great l y improved upon, with a small sacrifice in the
performance at the design incidence. At angles below 0°,
the performance could be somewhat ameliorated through an
improved design of the nose region. This .is required to
reduce the velocity spike at the lower surface leading
edge, which causes laminar separation. As a consequence
of the low design Reynolds number, the boundary layer
does not subsequently reattach, resulting in a rapid drag
rise at angles-of-attack below 0°.
The drag coefficient is calculated by the Squire-Young
formula (Ref. 19):
Hte+5 2
Cd = 2(9/c)te(u/u~)te (4.4)
This equation reveals the significance of the boundary
layer development, particularly in the pressure recovery
region, where the boundary layer grows most rapidly.
29
Figure 14 displays the deve10pment of the momentum
thickness on the designed section at the design incidence.
The 1inear rise of the momentum thickness in the recovery
region is characteristic of a velocity distribution with
a constant shape factor in this region. The difference
in the rate of increase of the momentum thickness between
the 1aminar and turbulent portions on both surfaces shows
the importance of de1aying transition. The large increase
in the momentum thickness at the trai1ing edge of the
upper surface, which is associated with a rise in the
shape factor (Fig. 13), is a re sult of the fact that the
trailing edge velocity is lower than the design value.
This has virtually no effect on the drag coefficient
because the lower velocity, which is raised to a power
of roughly 3.5 in the drag expression, compensates for
the increased momentum thickness at the trailing edge .
. For a given shape factor variation in the recovery
region, an increase in the design trailing edge velocity
results in higher velocities on the entire upper surface
and thus an increase in lift coefficient (Ref. 12). In
Reference 16 it is shown that the contribution of the
upper surface to the lift coefficient varies roughly
linearly with the trailing edge velocity for a specified
form of the upper surface velocity distribution. The
upper surface contribution to the drag coefficient also
30
rises with. an increase in the trailing edge velocity as
s'hown by equation 4.4. Therefore there exists a value of
the trailing edge velocity which maximizes the upper
surface contribution to the lift to drag ratio. However,
the increase in the velocities on the upper surface
associated with a rise in the trailing edge velocity
causes a reduction in the lower surface velocities for
a given thickness form, thus improving the section lift
to drag ratio. Hence the optimum value of the trailing
edge velocity can only be determined through a parametric
study in which a family of sections is designed, similar
to that used to find the optimum recovery point location.
The range of possible values is limited, however, especially
since the shape of the upper surface of the airfoil near
the trailing edge is determined by the nature of the
pressure recovery.
The lower surface velocity distribution displays the
desired characteristics at the design incidence (Fig. 12).
The velocity increases until x/c=O.89, where boundary
layer transition occurs. The section lift coefficient
could be improved by decreasing the airfoil thickness
aft of the location of maximum thickness, which is x/c=O.45.
This reduces the velocity on the aft portion of th.e lower
surface , causing earlier transition. At the design Reynolds
number, the resulting increase in drag was found to offset
31
the increase in lift.
Increasing the Reynolds number beyond the design
value of one million has two contradictory effects. The
transition point moves forwards but the skin friction
drag is reduced 50 the lift to drag ratio at the design
1ncidence is slightly increased. At a Reynolds number of
ten million, the lift to drag ratio is 192, although the
upper surface transition point is located at x/c=0.38,
compared to 183 at the design Reynolds number. At lower
Reynolds numbers, the transition point does not move aft
of x/c=0.7, since transition occurs by laminar separation
and reattachment when the laminar boundary layer is
exposed to the adverse pressure gradient at the start of
the recovery region. Therefore the increased rate of
growth of the boundary layer associated with a decrease
in the Reynolds number causes higher drag and eventually
flow separation, which is predicted at a Reynolds number
of 5 X 105 .
An extended transition ramp can be used to reduce the
sensitivity of a section to changes in Reynolds number. Ir transition occurs midway along the ramp at the design
Reynolds number, then the effect of changes in Reynolds
number on the boundary layer development are offset by
the movement of the transition point along the ramp. On
the designed section, the laminar boundary lay€r is
32
destaqilized over a very long region. However, transition
occurs just prior to the pressure recovery region at the
design incidence. Therefore this section is insensitive
to increases in Reynolds number but very sensitive to
decreases.
At the design Reynolds number, a forward shift in the
upper surface transition point results in a thicker boundary
layer at the recovery point and thus flow separation in
the pressure recovery region. Therefore this section
requires a smooth surface and cannot tolerate high free
stream turbulence levels, reflecting the initial assumptions
of the design study. On general aviation airplanes, the
type of wing construction of ten eliminates the possibility
of extended regions of laminar flow. For such applications
a similar design procedure can be used with a constant
velocity on the upper surface prior to pressure recovery,
assuming that transition occurs near the leading edge
regardless of the velocity gradient. As a consequence of
the resulting increased thickness of the boundary layer
at the recovery point, pressure recovery must be more
gradual to achieve a given shape factor variation. Thus
the performance of a section with a rough surface is
greatly reduced. For the lower surface, an aft portion
of reduced thickness, as discussed previously, is
advantageous since early transition is inevitable.
33
At high design Reynolds numbers, a substantial
positive velocity gradient is required to avoid transition.
Hence, for a given shape factor variation in the recovery
region, the recovery point location which maximizes the
lift to drag ratio moves forward as the Reynolds number
increases. An increase in the design Reynolds number has
a beneficial effect on the boundary layer development
and a detrimental errect on the stability of the laminar
boundary layer. Since the former effect has a greater
influence, the section lift to drag ratio which can be
achieved with given orr-design performance improves with
increasing design Reynolds number.
34
...
5." Conclusîons
The designed airfoil section is a reflection of the
design priorities which were initially established. At
the design Reynolds number, the section achieves a high
lift to drag ratio at the expense of the off-design
performance. The flow separation at the lower surface
leading edge at a relatively high positive lift coefficient
can be attributed to the design emphasis. However, the
separation of the flow on the upper surface is premature
even in light of the design priorities. The value of the
shape factor chosen for the pressure recovery region at
the design incidence was expected to allow a substantial
increase in angle-of-attack without separation. Early
separation is caused by a forward shift in the location
of transition, revealing the need for a short transition
ramp to restrict the movement of the transition point.
The importance of an appropriate recovery point location
in achieving high lift to drag ratios at low Reynolds
numbers is shown by comparing the performance of the
designed section with that of the Liebeck section, LI003.
Since the Liebeck section was designed for high lift, it
has a much earlier recovery point than the designed section
(See Figs. 8 and 18). At their design angles-of-attack,
35
the two sections exhibit approximately the same lift to
drag ratio Brit the designed airfoil has a considerably
lower shape factor in the pressure recovery region. Hence
it displays superior performance at reduced Reynolds
numbers. In addition, if the transition point were
adequately fixed, the designed section would have a
larger margin from stall and a higher maximum lift to
drag ratio, since this is obtained just prior to the
onset of stal 1 if the shape factor in the recovery region
is constant.
The use of the inverse turbulent boundary layer
equations rather than the Stratford distribution has
several advantages, including the ease with which desired
stall characteristics can be achieved. The shape factor
is an ideal link between the desired boundary layer
characteristics and the velocity distribution as it
quantifies the local proximity of both laminar and
turbulent boundary layers to separation. Because of their
simplicity, empirical criteria for boundary layer transition
and separation are weIl suited to the inverse design
approach. A simple transition criterion which accounts
for the effects of surface roughness and free-stream
turbulence would be helpful.
The use of a thickness distribution to determine an
initial lower surface velocity distribution led to a
36
favourahle final distribution. A more sophisticated
approach is required, however, to achieve an optimal
rorm for the lower surface, particularly at low lift
coefficients.
In addition, an analysis technique capable of handling
separated flows is needed to properly assess off-design
performance.
The basic design procedure described can be utilized
for the design of airfoil sections for many different
applications with various performance objectives, operating
conditions, and practical constraints. The inverse design
approach is extremely powerful, leading to high performance
airfoils which are not likely to be obtained by empirical
methods. As it begins with a set of boundary layer
characteristics, the inverse approach allows the design
priorities to be readily incorporated into the design.
The effect of different priorities can thus be easily
studied.
Several guidelines can be established for the design
of airfoils with high lift to drag ratios using an inverse
approach:
1. A constant shape factor variation in the pressure
recovery reg ion leads to high performance but a sudden
stall break. A more gradual stall break can be achieved
by increasing the shape factor towards the trailing edge.
37
2. A high. value of' the shape factor results in a high
lift to drag ratio at the design incidence but limited
performance at increased angles-of-attack and reduced
Reynolds numbers.
3. A short transition ramp leads to good performance
at off-design angles-of-attack while a longer ramp reduces
the sensitivity .of a section to changes in Reynolds number.
4. Optimum values of the recovery point and the trailing
edge velocity depend upon the operating conditions,
particularly the Reynolds number, and must be determined
through a study which accounts for the effect of these
parameters on the velocities on both surfaces.
Using these guidelines, an airfoil can be designed to
produce a high lift to drag ratio while achieving the
required off-design performance for a wide range of
operating conditions and practical constraints.
38
B. References
1. MeMasters, J. H. and Henderson, M. L., "Low-Speed
Single-Element Airfoil Synthesis", Science and
Techno1ogy of Low Speed and Motor1ess Flight, Part I,
NASA CP-2085, Mareh, 1979.
2. Liebeck, R. H., "A Class of Airfoi1s Designed for
High Lift in Incompressible Flow", J. Aircraft, Vol. 10,
No. 10, 1973, pp. 610-617.
3. Eppler, R., "Direkte Berechnung von Tragfluge1profilen
aus der Druckverteilung", Ing. Arch., Vol. 25, No. 1,
1957, pp. 32-57.
4. Cebeci, T., Mosinskis, G. J. and Smith, A. M. 0.,
"Calcu1ation of Separation Points in Incompressible
Turbulent Flows", J. Aircraft, Vol. 9, No. 9, 1972.
5. Stratford, B. S., "The Prediction of Separation of the
Turbulent Boundary Layer", J. of Fluid Mechanics,
Vol. 5, ~959, pp. 1-16.
6. Stratford, B. S., "An Experimenta1 Flow with Zero Skin
Friction Throughout its Region of Pressure Rise", J.
of Fluid Mechanics, Vol. 5, 1959, pp. 17-35.
7. Wortmann, F. X., "A Critical Review of the Physical
Aspects of Airfoil Design at Low Mach Numbers", Proc.
of Symposium on the Technology and Science of Motorless
Flight, M. I. T., Cambridge, Mass., 1972.
39
8. Cebeci, T., Mosinskis, G. J. and Smith, A. M.O.,
"Calculation of Viscous Drag and Turbulent Boundary
Layer Separation on Two-Dimensiona l and Axisymmetric
Bodies in Incompressible Flows", Rept. No. MDCJ0973-01,
Douglas Aircraft Co., Long Beach, California, 1970.
9. Henderson, M. L., "Inverse Boundary Layer Technique
for Airfoil Design", Advanced Technology Airfoil Research,
Vol. I, NASA CP-2045, Part 1, 1978, pp. 383-397.
10. Schlichting, H., Boundary Layer Theory, McGraw-Hill, 1960.
11. Cebeci, T. and Bradshaw, P., Momentum Transfer in
Boundary Layers, Hemisphere, 1977.
12. Liebeck, R. H. and Ormsbee, A. I., "Optimization of
Airfoils for Maximum Lift", J. of Aircraft, Vol. 7,
No. 5, 1970, pp. 409-416.
13. Egglestone, B. and Saville, H. G., "A Method for
Predicting the Aerodynamic Characteristics of Multi
.Element Airfoi1s with Viscous Attached Flow at Low
Mach Numbers", de Havilland Rept. DHC-PILP 78-3, 1978.
14. Zingg, D. W., Johnston, G. W. and Haasz, A. A.,
"Development of Computer Programs for Analysis and
Design of Two-Dimensional Hydrodynamic Sections",
Interna1 Rept., D. R. E. A., 1981.
15. Wortmann, F. X., "Airfoils with High Lift/Drag Ratio
at a Reynolds Number of about One Million", Proc. of
Symposium on the Technology and Science of Motorless
Flight, M. I. T., Cambridge, Mass., 1972.
40
16. Kennedy, J. L. and Marsden, D. J., "The Deve10pment
of High Lift, Single-Component Airfoi1 Sections",
Feb. 1979.
17. Abbot, I. H. and Von Doenhoff, A. E., Theory of Wing
Sections, Dover, 1959.
18. Wortmann, F. X., "Progress in the Design of Low Drag
Aerofoi1s", Boundary Layer and Flow Control, Vol. 2,
Pergamon, 1961.
19. Squire, H. B. and Young, A. D., "The Ca1cu1ation of
the Profile Drag of Aerofoi1s", A.R.C., R & M 1838,1937.
41
u/u
6
5. , J
2
I
.B·
.9
Fig. 1. Drag on a flat plate with different transition points (Ref. 7)
2.0
c 00 , , 1.0 "-
A Re = 1.5 X 105
B Re = 1.5 X 10 6
C Re = 1.5 X 107
......
O.O .... ____________ -A ______________ __
0.0 0.5 1.0 x/c
Fig. 2. Velocity gradients for transition at x/c = 0.7
42
C-3 6
H
Z 4
SIc x 103
1 Z
3
H
2
1
I I I I Upper Surface @ Q = 2°
I I
:: _-- 8xl04
} I I ... ' I I _ - - 6 Rn : 1 ___ ------- __ 3xl0 6 , ______ I _____ ---------- 20xlO
I tr
xlc
I1 trtr
. 5
Recovery Region •
1.0
Fig. 3. Shape factor variation on the NACA 633-018 (Ref. 1)
~llOOJ
(a)
/ /'
/ /
/
/' Upper
., SIc /
H
/ Surface
.5 1.0 xlc
3 6
H
2 4
1 2
Sic x 103
(b)
FX 74 - Cl6-140
laminar Bubble Predicted
xlc
Fig. 4. Boundary 1ayer characteristics of Liebeck L1003 andWortmann FX 74-CL6-140 airfoi1s (Ref. 1)
43
H
.0
.02
2.2
2.0
1.8
1.6 c,.
IA
1'----
i DESIGN . CTI;EORJ . .
Re-"Zxlcf
I Re.= 1.0 x 10' .04 L2 . 'fr', r.. c,. vs Cl
c..t./IC 0 1.0
- .02 ~ ~-=7=-C--:---.04 .8 . ' . L. C
u_ .. VI CI\
- .06 .-- \.. - .08 - c... vs C:> '"I'
! ,
o+---~---~~~-, __ ~ __ ~~~ .01 .02 .03 .04 .05 .05 .07
-4 -.2 <; 8 12 16 Co
Cl
Fig. 5. Experimenta1 lift, drag, and pitching
1000 moment data for Liebeck L1003 airfoi1 (Ref. 2)
tr
500
o--------~------~--__________ __ o 0.5 x/c 1.0
Fig. 6. Deve10pment of Rg ~nd Rgtr for a constan~
velocity distribution with Re = 2.0 X 10
44
2.0
. 1.0
o.o .......... ________________ .. ______ ~ 0.0 0.5 1.0
x/c
Fig. 7. Fami1y of upper surface velocity distributions
45
2.0/200 .
1.0/100
o 0.0
c1----
0.5 1.0
Fig. 8. Variation of lift coefficient and lift to drag ratio with recovery point location
46
!. . 0 .. : . . .. ! .... : ... "i .. .... .. ~ ... ..... ! .. . .... . . ! .. .. ..... : ..... . . . . ! . . . ...... ! ..•...• .. ! .. . .. ... . , . ...... .
i ....• ~ . ---:-~:-+--- ~----t - -'-+ - -t ---- - t - -T--:I~ - ---I - ~-·- , . ' : ~ : 1 : i
: . : ~ - _ •• 2 •••••••••.• .:. •........•. : ..•.•.••••• _ •.• _ •• _.: ..•••..•... _ •••••••.••• :.. . . ..•..... ~ .•..••.•.• . : . • ·_.·· •••• f,,: ... . . .. . . . ; .... . . _ .. . j .. _ ....... _ •........ .. ·······,······· .. _··_··-T-·· ... . . ! ~ i : : i _ooi, ·• • · •• · . 1·· ... . ·. ·,· ·· · · · · .· 1·· . j • •• :- ·UO .. : .. .. ,... ..... '::. i ..
i.. . ~.~ ..... ....... ....... .., ......... _ .......... L .... .. ... ~ ..
i;~ ~+~f:-:l+f-l-T+ -+::-~J~-Hifi~ I-F r······ ~ ~o ~ oo : I : 0~20 : I : O ~ llO : i ~ 0 ~ 60: I : 0 ~ 80~ i ~ 1~00 : L.: ... I~: . ! ..... . i ~. '1' ~~~~~I .. ·.: .. I.~ ·1 ~_:l~_X..I~.~L· .. · .. :.L .... · .. :.L .. : .. l· .. : .. · .. j.:.~~:J.·~~.:.L_:.:.L_:.: .L.~:.J: .. :.: .. :.L~.: .. J: .. :.: .. I.:_:.
Fig. 9. NACA 67-015 thickness form cambered to give design upper surface velocity distribution with x je = 0.7
r
1·······_·: ········· '--',-. -1---''':-'''-'' f"' ...... ~ ..... . .
j . ... : g .. ·· ·;····1· · '.j ' '.'! .... ! .... ~ .... ; .... ~ .. .. ~ ... ·1· ···:··· ·l····j·· ·· 1· ·· ·:··· · i· ·· · ~ ····i ···· j'" ., .. .. : .. .. t······_·:-······ i _
. . . j .... ~ .... !
........ _ .. __ ._ .... -
j .... :g. ·· · :··· · l··· · ·· ··· : · ··· ~ · · · · · · · · ·: ··· · l· · ·· ~ · ··· l····~····i""··· i ···· o ·· · ·j···· l · ··· :· · · · ~ ··· · :···· ~ · ·· ·:··· ·
·l:~::::=r~~~~~;-T--~ - • ·_-i::-I~i:~:ITEt~I~_~T:--~- _Er~~~~: : .. . . . ! .. .. : ... ! .... : .. , ... L ... : ... . :. . .. . · 0· ·· ·· ···· 0· ·· ·· ·· · · 0·· · ······ 0···· · . · · 0 . . ·.···.·
....... _.~._ ; :: i :: l I : l .......... ~ .......... . °OlOO: : 0~20 : Oi llO : : 0~60 : . : 0;80: . : l~OO :
! . ... . . . . .. , .i, .. • . . . .. . . ! ... .... , . • . . ·· X!",·. /C ·:. · ··' !",·.·· . .....•......... •. . ... .. ..• . .. ... ... , . .. . ..... , . . ... .. . . : : :. t.. ~..:::. i : . . : ........................ : .. .. ....... .:_ ........ 1 ........................ : ........... : .... ....... : .......... _ ........... L ........................ : ........... .: ........... t .......... _ .. _ ...... : ........................ : .................. ....... : ........... _ .......... : ..... .......... _ ... _ ..
Fig. 10. Velocity distribution on section shown above at ~= 3°
.. :.: .. :.~.:.:.~.: .. ~_~ .. :~.:.: .. :.·.L.:.:~.: .. · .. ~L~~~1 .. :~.:.~.:.: .. ~:.r··:·'· ···~·:·: .. ·.:·r .. :.: .. :.~.:.: .. :.J.· .. · ...... ~.·.: .. :.: .. l· .... · .. :.~:~ .. :~.I.: .. : ... ::.:~ ..... l. ........ ~._ .. .
.. . . : .... ~ ... . : .... ! .. . . : .... i ........ : ! ··.I. ... : ... t.--:· ... I.. ... ~ .... L ... ~.·.i .. ·-: .. .
. i . i . . ........ j ... ........ : .....•.•..• j ........... : ........... i ........... : ....... .
. . • i .. ·:~~~: ~:·· ·~·~--~··· ~i . --·~l·~ · ~r .... · .... i . . . · .... · .... ! .. .L.B.~.· 1 ~.o .! 1····: ··· · .... : .... ; .... : .... i .. .. : . . .. i .... : .. . . i ... . : .... ; .... : .... ; .... : .... ; . .. . : ... . ; . ... : .. .. ; . . . . : .... ; . ... : ...
.. ::.: . I . I : I : I : I : I : I ; I :
Fii·;·:o=r~:i 2:--T~;=~·T·~;·~~~---r--~ ~r:-·r· ·· l : ~~·~~ i··· · ~ .. .. i' .. . ~ .... j ... . ; .... j ... . ~ .... i···· ~ .. XV C ~ ·· . 'I'" . ~ ... . ~ .... ~ .... i' " . ~ .... i'" . ~ .... f···· ~ ... 'I' ... ~' . ' .
1 .. _ ..... : ........... !. .......... .:.. ... _L... ___ .:.._._ .. L ... _ . .:.. ___ L_~_ .. _i_ .... .... .:. ........... i. .•........ .: ........... L .......... .:. ........ _ .. L ......... .:._ ......... L .......... .:._ .. _.i ........... .:. ........... !. .......... : _ ...... .
Fig. 11. Designed airfoi1 section
.. : ........... :-........... ········· __ ······_·l·_·_·· ... ·· .. ······T·····_···:-····_-;---. - ··--i···········:-···········r···· .... _ ....................... _ ....................... _ ... ······f··········_···········:···········;·
I" ' .~ g . ·······T··· , .......... . : ... i .... : .... ! .. .. ;.- .. ; .. ..; .... i ... .; .... i .... · .. ··i· ... : .... ;. " ........ ~.:;::: . . ... .... ......... , ............ .. .. .... , ...... \ .......... _.. ..L······:···· .. ·····!· .. · .. ··: .. ·· .. · .. ·!······ .... :·· .. · .. ···r···· .. .. :······ .. ··T"···· .. ·~ ··· ·· .... ·i····· .. ·:· .. ····r ······ ... : ......... : i····· .. ·,·, .; ... ·· ! .. ·· · · .. ·!· .. ·· .. ·j" .. ····T· .. ·· ·· . . . . .. : ...... _ ... _ .. _.-1. ___ ... __ : ........ _._ ........... 1. .......... _ ..•...•.... : ........... _ ........... : ................... .. ;, ........... -............ ~::: .. · .......... 7.· .•• _._ •.• :,:" . . .......... .:. ..••. . .••.. . ; ........... _ ............... _ ... ···.··--T·----··-·..: .; " . ~" : j .
~ : : : : • • • • • • .! • • • • •• • • • • .~, • • • • ': • • • • '!, • • • • '. • • • • .!.'. • • I • •• •• . • . • • • . l ........ . ~, .......... I . • .. . • . !" .. • ...... ! .... : . ... ~ . . ... : ... .
j .. ~ ~ i : . . ~ ........... :.~. .. .. ··_···· ...... ·I .... · .. ·· .. .:. .. ···· .. ··~· .... · .. · .. _ .. · .. -··_~· .. ·_·_· .... -r .. _ .. ·· .. .:.· .. ··· .. · .. :·· ........ : .. _ ....... : ........................ (........... . ..... l·· .. · .. ·· .. .. · .... ··· .. ·1 .... ·· .. · .. :·· .... · .. ·"i'··· .. ·· .. · .. ···· ..... .
. : . ~ : : ;: 1 : l : 1 . : . : : 1 : ~ .
, u....;
j·~6 · L .. ~.~. j '-.:o
'" .:::J . . . ... i .. ; . . . . i,.:: . . . . . 1 . l ' . ~ . : . I . ............. _........... . ... : ........... ~ .... __ ... _ .. _ ..... ~, ........... _: ' ___ '-:',' _' ... ; : . . : '. l· . .' '. . · .. ·:·· .. ·· .. · .. :-·· .... ·····:····· .... ··:-.. · .. · .. · .. 1 .. ····· .. ··:-...... ·· .. ·r .... · .. · .. :- ·· .. ····· .. !· .. • · .. • .. ·:· .. • . .... ! . .. ... ..... : .. ........ : ........... :-......... .
. . . . i .... ~ . ..• i ..•. : . . •. i .... : .... i ... . : •... i .... : .... i .. .. : ... . i . . • . : ... .
. : . : .... . . , ......... , . . : : : • ; . ~ . . . 1 . : :.
........ : ........... , ........... : .......... ' ........... : ........... , ... _ .. - ... ··;···········:··········1··········:···········\···········:········· .. !······················1··········: ···········r·········:···········!·····················
. : .... ; . . : .... ~ ... . :- . . ! . . . : ... . : .... i ... .. ... . ! .... : .... ! •... : ... . ! ....... . T ... : .... :- ....... . ! . . . . . . . . .
.. : ....... ......... . ..... ...... L ..........•...•. .• i.. . .:.1.. ....... .. : .... ...... : ......... .. : .. ....... , .......... : ....... .... , .... ..... ; ......... , ....................... L. .............................. : ........ .
.. • . . . •..• !".... .j. : . . • • : .••• !:.· ..•. .•. · .... ~" .... : .... 1 .... : •...• , ..•..... • i . . .. . .... l .... : ... . !'''': o ' ..... .. · ·i· : . : : 0 ! ! .
i i. i . i i· .. " .: ...... ...... q ; 20 . ... i .. . ~}C : ··· . i .. ··:·· q r.6.0 : .. .. ! . . . : . . 9Lsq: ... '/"' .. :.' n.o.o;.- ..
. : ! : . .. .
. : .. ... .. .... _. ....... -:-=t ; 00:
i . ........ ! ....... .
. L .. .. ...... : ....... .... L .......... : ........... ! ... ....... ~ ........... L .............................. __ ................ : ........... : ........... : ........... 1. ....................... : ........... _ ........... L .......... _ ........... L ........ .. _ ..... ...... E ........... : .
Fig. 12. Velocity distribution on designed section at design incidence, 0(. = 3.50
48
: (1')
! · ···: o · :0
f-._._ .. _ ... .. -:. N , .
~ . ... : ... .
Fig. 13. Shape factor variation on designed section at design incidence
0 "-' ••••••• • • • •. I .: ~ . , r···_····~·o·
. ' . ': ~ . . . . . . • .. .. ••• ...•. I •.•••••• • I ....•
~~ :"';,;;::."-" : I :
: cj .... : --,::1' L .. x:,~.
. :. . ........ . , ....... . . , .. . .. . .. 1 · ··. l . 1
: '--: 0 .. , ........... .
~ ~ ._.T.---+---+ ---; - --, ----•
l .. ~ ~ . . ... ; .... ~ .. .. : .. .. ~ .. .. ; .... ~ . .. .. . .... '. _:~,; .· ... · .. · ... ·.·.: .. ,·.,·._· .. · ... ':i .. · . .. · . . · ... ·. _~ . . · .. · ... · .,·,.' .. ,~,: . . ' . . · . . . · .. · .. l.·",·.,·,',· .... , .... ! ........ ,., ... ,!, ..... , .......... :::,' ........ ----- , ------·r: r ......... -..... .. ... ; .......... ; ......... : .......... : ........... : ..... _.+ ..... - .. - - ",'
.,:".~, ........ _~: =. .;:: ...... :.,.J ..... : .. : .. ~ .... :.~ .'. ' ... : ...... ~ .. . . : . . . . . . , ... . ! . , .. i ' . , . : .... ! ' . .. : .. . ·l·· ': ............ î.. ... ,'" ....... :._: •. •• • •••.. ,' •• . :.--T· .. i' ...... : .. · .. ,· ........ : .......... I .......... ·: .... · ...... I ...... · .. : .. t·~· .... ,
;· .. ·: 6 .... : .... I .. .. : .. .. ! .. ·~·i .. . . .; , ·l .. ~~~lowèr : 0 : t :d surf'ace
.......... :.00 : 00 : : : 0' 20: 0: !t0 : ~ 0160: ol 80 ~ 1 I'ö'ü: ! . ... : .... i. . . . : .... i . .. . : .... i .... · . ~,.. . X'/C' .. . , .. .. ..... , ....... . i····:· ·· . j .. . ...... i •... : . .
: ~ 1 ~ :' . ,:.~: .':;. :.: . . :", . . . . . t .......... : ........... L .......... : ........... L ......... : ........... L .......... _ ...... __ , .. ~ ... ....... : .. ... ;.. .......... _ ........... : ........... .: ........... : ......................... : ................... ... ... :. .......... : ........... : ... .................... : ........... .
Fig. 14. Momentum thickness on designed section at design incidence
49
.. .... ..... ': .•. : ..... : ................ ~ ........... j ........... ~ ........ -~ ....... __ .· .. · .. · .. :· .. ·_··~~~··.r .. ·:·· .. ··~······: · ·.·O · ·. ·.·· ··:·~·:·· .. ·.··.T .. ·.-.... ·~·:· · .. ·:·· .. r .. ·:·· .. ·:·~·:·· .. ·:··.·;··.': ·.·:·~·.·· .. ·:··.··o·· .. ·.· ·.· -~· ·····: ·· .. ~·· .. ····.··.···:·· .. ·.·· .. ·1
' 0 ·· ·l··· ·: ····:··· ! 1 1 1. 1 j ,
,·· · · · ··· · ·f·~· ·· ·· ····~·· .. ·····j········ .. ~··········r····-~ .. _·· .... ·· ...... ~~ ........ ; .......... -........... ! .......... ~ ....... ··T··········-·········i .......... ~ ........... ! ...................... .' ....................... , ........... : .......... ~
;~---~-----: •••• : ••• ~~=:-~~i---:-}-:-::-::_:I~-: -[~::;::~: ~-.. .. . i ~ . ~ i l 1 1 l ~ j ~ ~ ........... ~.:;:::. . ........ ······ .... i·· .. ····· .. ~··· ······1---.. ··~--+··_:._-·+········ .. : ......... -+ .......... : .... ······,···········:···········i··········· .... -+ ...................... j ..... .. ................ l ...................... . i .. j • • • • •• ••• j • • . •• . • •• , • •• • . • .••• ;': •••••••.. . ; . •• ••• ••• i ....•.... i ......... 1 • • •• • • • i". : . . . . . . . . . l: j': ~
t····CL····· .. ···· ... ······-······ .. ·· ·~········· · ·- · .. ····· ·4········---···· .... ~ ........ --.-...... j .......... : .......... ! ........... -......... + ......... -........... ! ........... -........... ~ ................. .
~ ..... ·3 .. ~. · '0 ;. ::J
, . ~ ........... _ .......... .
j ••
' 0 : ;:1'
~ ........... _ ...... . : 0
. ....... ~ .......... + ......... -.......... ~.-.... -.. _ .... -... , ...... -.----.... ~··········-··········r·········· · ··· ··· ····1·· · ···· ·· ··: ·· ········ · 1· ··· ··· · ·· ·:· · .. ··· · · · · ~ ··· ········7·· ··· · ····~····· · ·····-··· · ·······l·· · ........ -.... .
o . .. . ! . .. ...... ! . .. - . ~ .. ······i ·········i ......... ! ..
...................... : .................... _.-: ........... -·_····~···········_··_· · ···i·· ........ _ ........... : ..... ... _._ ........... ~ .. ........ ·_ · ·_·······~··········· _· · ·········i·· · ········_····· ...... ;
··i· · ·· ·· ... j. · ·· · l ··· ···· ··j·· . ... . .. ! . . .
...••..• _ ...•..•.••. ~ .•••.•... _ •.•••. .... : ........ _._ .... . _.~ ..•...•..• _ ....................... _." ....... j .. ......... _ ........... j ........... _ .. ......... j ........... _ ........... : ........... _ ........... ~ ........ ... _-_ ....... ~ .......... _ .......... .
. j · i .... .. , . .. ···· ··i.·· ···· · ·,·· ···I . ·· .. . · ·· j· · . . . ! . . . . . . . . . ~ . . . , .
.......... _ ...................... _ ....... " .. : ... ··· .. ··_· ...... ·· .. :· .. · .. • .. ··_ .. _·_· .. j_· .. ·_ .. __ ·····_·i ...... ·· .. ·_ ....... . .......... ~ ........... _ ........... : ........... _ ........... : ........... _ .. ·· .. ·· .. ·l .. · .. · .... ·_· .. · .. · .... :·· .. · .... ··_·· .. ···· .. ·
· 0 : 0
. ~ . , .
. . ! . . . . . .. . ~
., ........... -...... -+----.:...-.;.-.....:.-.--+---'-..----..:-+-~-'----+---+---+--_+---f ........... -.......... . ai ilO : Ol60: ·· ·XilC:·· ··o
• •• • ••••• ! . 0; 20 : . I • . .•.. .. . ! . . Oi 80 :
·t·········,· ·····1 ·
,:._ .... .... _ .......... :_ ......... _ ........... :.. .......... _ ......... ............. _ .. _ ......... . . .. _ ...................... _ ....................... _ ................... .... _ ......... J .. " ....... _ .. ......... l ........... _ ..... .... _ ......... _ .......... .
Fig. 15. Velocity distribution on designed section at ~ = 1.00
50
0"': Design point
1.5 1.5 1.5
Cl / ....•. Cl .f····· Cl
1.0 1.0 1.5
•• •••• ... I I •• •• •• --. •• .
\.T1 0.5" 0.51- 0.5 ~
0.0 • , . 0.0 0.0 0.0 5.00 10.00 0.0 0.005 0.010 0.015 0.0 -0.2
oe: Cd Cm
Fig. 16. Ca1cu1ated lift, drag, and pitching moment characteristics of the deslgned airfol1 at Re = 106
·0
r·:J;-~·J~~T"~L:):-I~=8JITI:i:.I.""· .••. -.••. :.i ... · .••.••. ~:· . • ·.-.·.~.· •. :i .. :: .. ~. ~· .• ·-.. -.·:.~.·.···.-:.-.:· _".·.··~. · :.·:.J.l.-·.-:.·~.~ •.•.. : ...... ~."" .. ~ .... : ..... ... : .... l' · .. :·· ·· ~"":'" '1' " :' .:. :. :::::' ;'. .: l . ! . 1 . I . ~
! . . . . : ....
t}· I~i=Id~~ •• i-:FJ-I::~±lT:f+Jtq~l;fIJ ~--~. --.:-~-I~::~I:-J::t=1J:I::i~:.l~.:T~~:~g-:. -4I:.Ii::ç, .:_ ... _._.l.~ : . ~; '~i I i' i . I : I . I .......... ~ ....... _.: j :9roo: : Oi 20 : : Oi llO: : 01 60 : : olaD: : L OD: L.·.~~.l.:_·.~ . ..J.·_~·~~.·.: .. : .. J: .. : .... :.L· ..... .J..· ....... :L.: .. : .. J.~_i_·_~.r~ .. ~L.:.: .. :L .. ·.: .... L .. ~.J. ... · .. :.:.:.· .. :. ·..1..· .. : ...... :.:.· .. ·.: .. : .... .. : ... 1 ........ :.L ... : .. :.~.:.· .. : .. :.L: ... ~.~.:~:.~~ Fig. 17. Velocity dist r ibution on designed section at ~ = 40
-3 Rn • 1 x 106
C; • 1.8
Cp -2
-1
o+-------------~--~~~--~
___ .-:.:-_--;---r . 0 xtc
+1
Fig. 18. Velocity distribution on Liebeck L1003 airfoi1 at design incidence (Ref. 2)
52
· Appendix A. Thwaites' Method for the Laminar Boundary
. Layer Gal c ulati on
The computer program which determines the velocity
gradient required for a given transition point (Appendix C)
utilizes Thwaites' integral method to perform the laminar
boundary layer calculation.
The momentum thickness is calculated from the following
equation:
2 (~) Re c
This allows the determination of the parameter A :
g du
From known solutions of the boundary 1ayer equations,
(A.1)
(A.2)
Thwaites found the following reasonab1y valid universal
functions l(À) and H(À):
2
1 1 = 0.22 + 1.57À - 1.SA
3.75-\ + 5.24A2 O!: À~ 0.1
H = 2.61 -
O.OlSÀ (A. 3)
1 = 0.22 + 1.402À + O.107+,À
0.0731 + 2 088 -0.1~).~0
H = 0.14+,\ .
53
The shape factor is given by the function H(A). The skin
friction coefficient is found from:
(A.4)
54
Appendix B. Inverse Turbulent Boundary Layer Program
C INVERSE TURaULENT 8JUNCARY LAYER PRUGRAM C. C. CALCuLATCS VELOCITY ~ISTRI D UTION CORRESPONDING Ta A GIVE~ C SHAPE FACT:JR VAr.:IATICJ'-' AFTER TRAN3ITION
1 IMPLICIT REAL:!r8 (A-H.O-Z) 2 :,HI-1cNSION C(24),W(Z,C;),Y(2),YP(2) 3 EX H:.RNAL FCN 4 COMMON "i(JOO),DHDX(lOO),RE.I 5 IT=1 6 N\'.=2 7 TjL~.OOI 8 ITMAX=20 9 TOl2~.OOOl
C. READ I N NUlv\ijER CF PCINTS. L:JCATIUN Of TRANSITION, I:'IIITIA_ GUESS C. AT ~TA~TING VELOCITY. AND TRAILING EDGE VELOCITY
10 RE:AD(5.100) NT.XR.URI,UTE 11 100 F~RMAT(14,3F8.0)
C. RCAD IN CHCKD PEYNOLiJS I\O~LlER 12 P CAu(5,110) RE 13 110 FUK,.l ATCFB.O)
C ~EAD l~ H VHRIATION (AND DH/DS) 14 REAi)(5.120)CHCI),I=I,NT) 15 PCA l> (!:>,120)CDHOXCI),I=I.NT) 16 120 F 0 R:-lA T ( 1 OF a. 0) 17 ~RITE(6,230) RE 18 230 FORMAT(OI5.3)
C CA:.... CU LAT E r< T ii ETA F 0 R TRA hl SI TI 0 NAT X R 19 1 HXl= RE*URI*XP 20 RTH! =1.174*( 1. +22400 ./RX I) *RXI**0.46 2 1 H R IJ E C (). 2 4 0) R Xl, f.! T H I 22 240 FUR~AT(2DlS.6)
L CA~CULATE INITIAL THETA 23 THI=RTHI/~C/URI 24 YCl)=URI 25 X= XR 26 V(2)=THI 27 .... r:l I lt:::( 6,200)X,URI .TH I,HC 1) ,DHDX Cl) 28 200 FJR~ATC5FI0.5)
C INTEGRATE USING A RUNGE KUTTA-METHOD 29 :) 0 2 I =1 , NT 30 XEND=X+.0125 31 IND=l 32 CALL DVEr<KC2,FCN,X,Y,XEND.TOL,IND,C.NW,~.IER) 33 X=XCND 34 IFClNO.LT.O.O~.IER.GT.O) GO TO 3 35 -'.;.(ITlC6.2JO) X.YCl).V(2),H(2).DHDX(I) 3ó 2 CONTINUE
C AOJUST INITIAL VELOCITY AND ITERATE UNTIL DESIRED TRAILI'-'G C EDGE VFLUCITY IS ACHIEVEO
37 IF(DABSCY(1)-JTE).LT.TOL2} STOP 38 LJt<I=URI+UTE-Y( 1) 39 lT=I1+1 40 IFCIT.GT.IP,.,AX) GiJ TO 4
55
3 210
GO TG 1 w RIT E ( b. 2 1 0 ) I ND • IER FJRMATC5X,I4.14} ST OP l'IRITCC6.220) lTMAX
41 42 43 44 ' 45 46 47 48
4 220 F 0 Ri·1AT (5X. • ITt.1 AX EXC EEL>EO' • 14)
STOP E\I 0
49 S0aruUTINE FCNCN.X.Y.YP) 50 lMPLICIT RC/ll*B (A-H.O-Z) 51 DIME:'NSlûN Y(:-~).Y?(2) 52 COMMON H(100).DH~X(lOO).RE.I 53 E=2.71H281B28 · 54 RT H= RE * Y ( 2 )
L LUJ~IG-TILLMAN EQUATI3N 5~ TJ=O.123*~TH.*'-.268)*10.**(-.67B*H(I»
C REARRANGEC FD~M ~F GARNEP'S EQUATION 56 Y P ( 1 ) = -L) H ~ X ( I ) *' Y ( 1 ) / E* * ( 5. >I: ( H ( I ) - 1 • 4 ) ) - 0 • 0 1 35* ( H Cl) - 1 • 4} * Y ( 1 )
2/~TH**(1./6.)/Y(2) C MUMLNTUM EaUATIO~
57 Y=>(2)=TU-(rlCI)+2.>*Y(2)*YPCl )/Y(l) 58 RETURN 59 CND
56
.,
. Appendix D .COOrdinates Of Designed Airf'oil
x/c y/cupper y/clower
• 1. 000 0.00000 0.00000
0.975 0.00959 -0.00030
0.950 0.01872 -0.00059
0.925 0.02737 -0.00089
0.900 0.03603 -0.00119
0.875 0.04485 -0.00148
0.850 0.05395 -0.00178
0.825 0.06337 -0.00207
0.800 0.07313 -0.00237
0.775 0.08319 -0.00267
0.750 0.09349 -0.00296
0.725 0.10360 -0.00326
0.700 0.11470 -0.00356
0.675 0.12080 -0.00385
0.650 0.12710 -0.00415
0.600 0.13496 -0.00474
0.550 0.14010 -0.00533
0.500 0.14287 -0.00593
0.450 0.14348 -0.00652
0.400 0.14206 -0.00700
58
x/c Y/Cupper y/clower
0.350 0.13863 -0.00740
0.300 0.13310 -0.00770
0.250 0.12525 -0.00800 •
0.200 0.11500 -0.00830
0.150 0.10140 -0.00860
0.100 0.08400 -0.00890
0.075 0.07100 -0.00900
0.050 0.05700 -0.00907
0.025 0.03800 -0.00798
0.0125 0.02700 -0.00633
0.0075 0.01800 -0.00500
0.005 0.01300 -0.00400
0.000 0.00000 0.00000
59
..
UTIAS Technical Note No. 245
Institute for Aerospace Studies, University of Toronto (UTlAS) 4925 Dufferin Street, Downsview, Ontario, Canada, M3H 5T6
AN APPROACH TO THE DESIGN OF AIRFOILS WITH HIGH LIFT TOORAG RATIOS
Zingg, David Walter
1. Subsonic airfoil design
1. Zingg, David Walter
2. Two-dimensional inverse/design
II. UTIAS Teehnical Note No. 245
~ 3. Panel methods
A procedure for the design of low-speed, single-element airfoils with high lift to drag ratios is presented. The procedure us es an inverse approach which proceeds from a set of desirabie boundary layer characteristics, which are determined from the performance objectives, to a velocity distribution and, finally, to an airfoil shape . The boundary layer shape factor, which is a measure of the neamess of the boundary layer to separation, is used to define the boundary layer eharacteristics in the upper surface pressure recovery region. The turbulent boundary layer equations are solved in inverted form to calculate the velocity distribution which corresponds to a shape factor variation . The veloeities on the upper surface prior 'to pressure recovery are chosen to cause boundary layer transit ion at a desired location. The lower surface velocity distribution is selected to satisfy the requirement that a closed, non-reentrant shape which meets the specified structural requirements resul ts . The design procedure described can be utilized for the design of high performance airfoils for a variety of performance objectives. operating conditions, and practical constraints.
Using this procedure, an airfoil has been designed which achieves a lift to drag ratio of 180 at a lift coefficient of 1.2 and a Reynolds number of one million. Through this design study, a set of guidelines for the design of airfoils with high lift to drag ratios is established.
Available copies of this report are limited. Return this card to UTIAS, if you require a copy.
UTIAS Technical Note No. 245
Institute for Aerospace Studies, University of Toronto (UTlAS) 4925 Dufferin Street, Downsview, Ontario, Canada, M3H 5T6
AN APPROACH TO THE DESIGN OF AIRFOILS WITH HIGH LIFT TOORAG RATIOS
Zingg, David Wal ter
1. Subsonic airfoil design 2. Two-dimensional inverse/design
I. Zingg, David Walter II. UTIAS Technical Note No. 245
~ 3. Panel methods
A procedure for the design of low-speed, single-element airfoils with high lift to drag ratios is presented. The procedure uses an inverse approach which proceeds from a set of desirable boundary layer characteristics, which are determined from the performance objectives, to a velocity distribution and, finally, to an airfoil shape. The boundary layer shape factor, which is a measure of the neamess of the boundary layer to separation, is used to define the boundary layer characteristics in the upper surface pressure recovery region. The turbulent boundary layer equations are solved in inverted f arm to calculate the velocity distribution which corresponds to a shape factor variation. The ve loeit i es on the upper surface prior to pressure recovery are chosen to cause boundary layer trans i tion at 8
desired location. The lower surface velocity distribution is selected to satisfy the requirement that a closed, non-reentrant shape which meets the specified structural requirements resul ts . The design procedure described can be utilized for the design of high performance airfoils for a variety of performance objectives. operating conditions, and practical constraints .
Using this procedure, an airfoil has been designed which achieves a lift to drag ratio of 180 at a lift coefficient of 1. 2 and a Reynolds number of one million. Through this design study, a set of guidelines for the design of airfoils with high lift to drag ratios is established.
Available copies of this report are limited: Return this card to UTIAS, if you require a copy.
UTIAS Technical Note No. 245
Institute for Aerospace Studies, University of Toronto (UTlAS) 4925 Dufferin Street, Downsview, Ontario, Canada, M3H 5T6
AN APPROACH 1'0 THE DESIGN OF AIRFOILS WITH HIGH LIFT TO DRAG RATlOS
Zingg, David walter
1. Subsonic airfoil design 2. Two-dimensional inverse/design
I . Zingg. David Walter Ir. UTIAS Technical Note No. 245
~ 3 . Panel methOds
A procedure for the design of low-speed, single-element airfoils with high lift to drng ratios is presented . The procedure uses an inverse approach which proceeds from a set of desirabic boundary layer characteristics, which are determined from the performance objectives, to a velocity distribution and, finally, to an airfoil shape. The boundary layer shape factor, which is a measure of the neamess of the boundary layer to separation, is used to define the boundary layer characteristics in the upper surface pressure recovery region. The turbulent boundary layer equations are solved in inverted form to calculate the velocity distribution which corresponds to a shape factor variation. The veloeities on the upper surface prior 'to pressure recovery are chosen to cause boundary layer transition at a desired location. The lower surface velocity distribution is selected to satisfy the requirement that a closed, non-reentrant shape which meets the specified structural requirements resul ts. The design procedure described can be utilized for the design of high performance airfoils for a variety of performance objectives. operating conditions, and practical constraints .
Using this procedure, an airfoil has been designed which achieves a lift to drag ratio of 180 at a lift coefficient of 1.2 and a Reynolds number of one million . Through this design study, a set of guidelines for the design of airfoils with high lift to drag ratios is established.
Available copies of th is report are limited. Return this card to UTIAS, if you require a copy.
UTIAS Technical Note No. 245
Institute for Aerospace Studies, University of Toronto (UTlAS) 4925 Dufferin Street, Downsview, Ontario, Canada, M3H 5T6
AN APPROACH TO THE DESIGN OF AIRFOILS WITH HIGH LIFT TO DRAG RATlOS
Zingg, David Walter
1. Subsonic airfoil design
I. Zingg, David Wnlter
2. Two-dimensional inverse/design
I I. UTIAS Technical Note No. 245
~ 3. Panel methods
A procedure for the design of low-speed, single-element airfoils with high lift to drag ratios is presented. The procedure uses an inverse approach which proceeds from a set of desirable boundary layer characteristics, which are determined from the performance objectives, to a velocity distribution and, finally, to an airfoil shape. The boundary layer shape factor, which is a measure of the neamess of the boundary layer to separation, is used to define the boundary layer characteristics in the upper surface pressure recovery reg ion . The turbulent boundary layer equations are sol ved in inverted form to calculate the velocity distribution which corresponds to a shape factor variation. The veloeities on the upper surface prior 'to pressure recovery are chosen to cause boundary layer transition at a desired location . The lower surface velocity distribution is selected to satisfy the requi rement that a closed, non-reentrant shape which meets the specified structural requirements results. The design procedure described can be utilized for the design of high performance airfoils for a variety of performance obj ectives, operating conditions • and practical constraints .
Using this procedure, an airfoil has been designed which achieves a lift to drag ratio of 180 at a lift coefficient of 1. 2 and a Reynolds number of one million . Through this design study, a set of guidelines for the design of airfoils with high lift to drag ratios is established.
Available copies of th is report are limited. Return this card to UTIAS, if you require a copy.