AIRFOIL SHAPE FOR ARRAYS OF AIRFOILS

AUBE AERODYNAMIQUE POUR ARROIS D'AUBES

English Abstract

Abstract
A flow directing assembly 14 having an airfoil
section or shape 28 of the type adapted for use in an
axial flow gas turbine engine is disclosed. The
cambered meanline MCL of the airfoil shape is formed of
a front circular arc FA and a rear circular arc RA. A
thickness distribution TD is applied to the meanline
to form the convex suction surface 20 and the concave
pressure surface 22. The airfoil section exhibits good
aerodynamic performance as compared with an equivalent
circular arc airfoil in a transonic flow field. A
method for making the airfoil shape is disclosed. The
method includes the steps of: forming a cambered mean-
line of two circular arcs; forming a thickness distri-
bution about the conical chord line Bt; and applying
thickness distribution to the cambered meanline such
that a portion of the suction surface is stretched and
a portion of the pressure surface is compressed.

Claims

- 30 -
The embodiments of the invention in which an exclusive
property or privilege is claimed are defined as follows:
1. A rotor blade having one or more airfoil sections,
each airfoil section being one of a plurality of airfoil
sections which are circumferentially spaced a distance
tau (?) apart about a rotor axis, each airfoil section
having an inlet metal angle .beta.?, a total camber angle .theta.?,
an alpha chord angle .alpha.ch, a maximum thickness tmax, a
leading edge, a trailing edge, a tangent line TL passing
through the leading edge tangent to the path of rotation,
a front chord line Bf of length bf, and a conical chord
line Bt of length bt wherein the values of .beta.?, .theta.?, ?,
bt, the maximum thickness of the airfoil section tmax,
are known, the rotor blade having one or more airfoil
section geometries determined by the method steps of:

- 31 -
A, establishing a cambered meanline having
a concave side and a convex side and having a
first arc, a second arc and a transition point
TP between the first arc and the second arc,
the first arc being tangent to the second arc
at said transition point TP by
Aa. determining an initial value for the
alpha chord angle (.alpha.chi) which is equal
to the sum of the inlet metal angle (.beta.?)
and one-half of the total camber angle
<IMG>, (.alpha.chi = .beta.? + <IMG>,
Ab. setting the value of the alpha chord
angle .alpha.chi (.alpha.ch = .alpha.chi),
Ac. determining a distance ? from the
tangent line TL to the first covered section
as measured along the conical chord line Bt,
the distance ? being equal to the distance
tau ? multiplied by the quantity the sine
of the angle ninety degrees minus the
alpha chord angle (? = ??sin(90-.alpha.ch),
Ad. determining the normalized distance
Lfcs to the first covered section by
dividing the distance ? by the distance
bt.
Ae. obtaining the ratio of the length
bf of the front chord line Bf to the
length bt of the conical chord line Bt
(bf/bt) and the ratio of the front camber
angle (.theta.?) to the total camber angle .theta.?
(.theta.?/.theta.?) as a function of the value Lfcs
of the normalized distance to the first
covered section,

- 32 -
Af. establishing the location of the first
arc such that the arc passes through the
leading edge using the values known (bt,
.theta.?, .beta.?) and the value found in step Ae. for
bf and .theta.f,
Ag. establishing the location of the second
arc such that the arc passes through the
trailing edge using the values known (bt,
.theta.?, .beta.?) and values found in step Ae, for
bf, .theta.?,
Ah. establishing a conical chord line Bt
extending between the leading edge and the
trailing edge,
Ai. determining the actual alpha chord
angle .alpha.cha for the cambered meanline,
Aj. determining the difference E between
the actual alpha chord angle .alpha.cha and the
alpha chord angle .alpha.ch used to calculate the
normalized location Lfcs by subtracting
.alpha.ch from .alpha.cha (E = .alpha.cha-.alpha.ch),
Ak. proceeding to step B if the absolute
value of E is less than the predetermined
value e (¦E¦<e) and proceeding to step Am
if the absolute value of E is greater than
or equal to the predetermined value
e (¦E¦?e),
Al. setting the value of the alpha chord
angle .alpha.ch equal to the value .alpha.cha
(.alpha.ch = .alpha.cha),
Am. repeating steps Ac through Aj;
B. establishing a thickness distribution TD
having a line spaced a distance Tzn from the
conical chord line Bt at any point zn, the point
zn being spaced a distance Lan from the leading
edge on the conical chord line Bt, the distance

- 33 -
Tzn being measured along a line Zn perpendicular
to the conical chord line Bt,
C. superimposing the thickness distribution
on the cambered meanline by
Ca. establishing a plurality of points
zn', each point zn' being at the inter-
section of the line Zn and the cambered
meanline,
Cb. establishing a line Z'n perpendicular
to the cambered meanline at each point zn',
Cc, establishing a point zn" at a distance
Tzn as measured along the line Z'n from
the convex side of the cambered meanline
at each point zn' and a point zn''' at a
distance Tzn as measured along the line Z'n
from the concave side of the cambered mean-
line at each point zn',
Cd. establishing a concave surface passing
through the leading edge and the trailing
edge and through all points zn",
Ce. establishing a convex surface passing
through the leading edge and the trailing
edge and through all points zn'''.

- 34 -
2. A rotor blade having one or more airfoil
sections, each airfoil section being one of a plurality
of airfoil sections which are circumferentially spaced
a distance tau (?) apart about a rotor axis each of
the airfoil sections having an inlet metal angle .beta.?,
a total camber angle .theta.?, an alpha chord angle .alpha.ch, a
maximum thickness tmax, a leading edge, a trailing
edge, a tangent line TL passing through the leading edge
tangent to the path of rotation, a front chord line Bf
of length bf, a conical chord line Bt of length bt wherein
the values of .beta.?, .theta.?, ?, bt, the maximum thickness of the
airfoil section tmaX, are known, the rotor blade having
one or more airfoil section geometries determined by the
method steps of:
A. establishing a cambered meanline having a
concave side and a convex side and having a
first arc, a second arc and a transition point
TP between the first arc and the second arc,
the first arc being tangent to the second arc
at said transition point TP by
Aa. determining an initial value for the
alpha chord angle (.alpha.chi) which is equal to
the sum of the inlet metal angle (.beta.?) and
one-half of the total camber angle <IMG>
(.alpha.chi = .beta.? + <IMG>,
Ab. setting the value of the alpha chord
angle .alpha.chi (.alpha.ch = .alpha.chi),
Ac. determining a distance ? from the
tangent line TL to the first covered section
as measured along the conical chord line Bt,
the distance ? being equal to the distance
tau ? multiplied by the quantity the sine
of the angle ninety degrees minus the alpha
chord angle (? = ? sin(90-.alpha.ch),

- 35 -
Ad. determing the normalized distance Lfcs
to the first covered section by dividing
the distance ? by the distance bt,
Ae. obtaining the ratio of the length bf
of the front chord Bf to the length bt
of the conical chord line Bt (bf/bt) and
the ratio of the front camber angle (.theta.?)
to the total camber angle .theta.? (.theta.?/.theta.?)
at the value Lfcs of the normalized distance
to the first covered section,
Af. establishing the location of the first
arc such that the arc passes through the
leading edge using the values known
(bt, .theta.?, .beta.?) and the value found in step
Ae. for bf and .theta.?,
Ag. establishing the location of the
second arc such that the arc passes through
the trailing edge using the values known
(bt, .theta.?, .beta.?, and values found in step Ae,
for bf, .theta.?,
Ah. establishing a conical chord line Bt
extending between the leading edge and the
trailing edge,
Ai. determining the actual alpha chord
angle .alpha.cha for the cambered meanline,
Aj. determining the difference E between
the actual alpha chord angle .alpha.cha and the
alpha chord angle .alpha.ch used to calculate
the normalized location Lfcs by subtract-
ing .alpha.ch from .alpha.cha (E = .alpha.cha-.alpha.ch),
Ak. proceeding to step B if the absolute
value of E is less than the predetermined
value e (¦E¦<e) and proceeding to step
Am. if the absolute value of E is greater
than or equal to the predetermined value
e (¦E¦?e),

- 36 -
Al. setting the value of the alpha chord
angle .alpha.ch equal to the value .alpha.cha
(.alpha.ch = .alpha.cha),
Am. repeating steps Ac through Aj,
B. establishing a thickness distribution TD
formed of two parts each part being disposed
about the conical chord line Bt, each part
having a line spaced Tzn from the conical chord
line Bt at any point zn, the point zn being
spaced a distance Lan from the leading edge
on the conical chord line Bt, the distance
Tzn being measured along a line Zn perpendi-
cular to the conical chord line Bt, the line
of the first part being TD1 and the line of
the second part being TD2,
Ba. the line of the first part TD1 being
established by
Bal. determining the distance loc mt
along the conical chord line to the
location TMAX of maximum thickness
tmax by determining the ratio <IMG>
as a function of the value Lfcs
of the normalized distance to the
first covered section,
Ba2. superimposing on the conical
chord line Bt a circle Tmax having a
center on the conical chord line a
distance equal to loc mt from point
A and a radius Rtmax equal to one-
half of the maximum thickness tmax
of the airfoil section (Rtmax = <IMG>),
Ba3. establishing on the conical
chord line Bt a leading edge radius
circle having a radius Rler and a
center on Bt a distance equal to Rler

- 37 -
from the leading edge and intersect-
ing the leading edge at a point A, the
radius Rler being equal to a first
constant k multiplied by the maximum
thickness tmax (Rler = k , tmax),
Ba4. establishing a line Q perpendi-
cular to the conical chord line Bt
at a point which is a distance bf
(Lan = bf) from the leading edge,
Ba5. establishing a line F having
a radius of curvature Rf which is
tangent to the leading edge circle at
a point f?, tangent to the circle
Tmax and which intersects the line Q
at a point fq,
Ba6. establishing a line P perpendi-
cular to the conical chord line Bt at
a point which is a distance Lan equal
to a second constant k2 multiplied by
the length bt of the conical chord
line (Lan = k2 ? bt) from the leading
edge and which intersects the line F
at a point fe,
Ba7. passing the line TD1 of the
first part through the points A, fe
and fq such that the line of the first
part is tangent to the leading edge
radius circle at point A, tangent to
the line F at point fe and coincident
with line F between the points fe and
fq,
Bb. the line of the second part TD2 being
established by

- 38 -
Bb1. determining the quantity TERG
as a function of the value Lfcs of the
normalized distance to the first
covered section and determining the
radius Rter which is equal to the
quantity TERG multiplied by a third
constant k3 and by tmax
(Rter = TERG '.463?tmax),
Bb2. establishing on the conical
chord line Bt a trailing edge radius
cicle having a radius Rter and a
center on Bt spaced a distance equal
to Rter from the trailing edge and
intersecting the trailing edge at
a point C,
Bb3. establishing a line G having a
radius of curvature Rg which is
tangent to the trailing edge radius
circle at a point gt and which is
tangent to the line F at the point
fq,
Bb4. passing the line of the second
part TD2 through the points C, gt
and fq, such that the line of the
second part is coincident with the
trailing edge radius circle between
the points C and gt and coincident
with the line G between the points
gt and fq,
C. superimposing the thickness distribution on
the cambered meanline by
Ca. establishing a plurality of points
zn', each point zn' being at the inter-
section of the line Zn and the cambered
meanline,

- 39 -
Cb. establishing a line Z'n perpendicular
to the cambered meanline at each point
zn',
Cc. establishing a point zn" at a distance
Tzn as measured along the line Z'n from
the convex side of the cambered meanline
at each point zn' and a point zn''' at a
distance Tzn as measured along the line
Z'n from the concave side of the cambered
meanline at each point zn',
Cd. establishing a concave surface passing
through the leading cage and the trailing
edge and through all points zn",
Ce. establishing a convex surface passing
through the leading edge and the trailing
edge and through all points zn''',
wherein the thickness distribution is stretched
chordwisely on the convex side and compressed on the con-
cave side to form an airfoil section having desirable
separation characteristics in a transonic aerodynamic
flow field.

Description

1 ~ 7~575
This invention relates to axial flow rotary
machines and particularly to transonic airfoils for use
in such a machine.
Axial flow rotary machines typically have arrays
of airfoils extending across a flow path for working medium
gases. The airfoils of each array receive work from the
working medium gases or do work on the working medium gases
by turning the flow. As the gases pass through the array,
the gases may experience shock waves and separation of the
boundary layer of the gases from adjacent airfoil surfaces.
These pheomena cause aerodynamic losses. The losses limit
the stage efficiency of the airfoils. The losses are of
particular concern in a transonic flow field, i.e. any
flow field which contains regions of subsonic and super-
sonic local velocity in juxtaposition A discussion of
this subject is available in Wu and Moulden "A Survey of
Transonic Aerodynamics'`, AIAA Paper No. 76-326, presented
at the AIAA Ninth Fluid and Plasma Dynamics Conference,
San Diego, California, 1976.
One way to reduce the losses in a transonic
flow field is to optimize the contour of the airfoil.
This approach was emphasized during the last two decades.
A result of such work found expression in U. S. Patent No.
3,952,971 to Whitcomb entitled ~Airfoil Shape for Flight
at Subsonic Speeds~. The Whitcomb patent deals with an
isolated airfoil having no internal or guided flow.
However, this patent is an example of an improvement in
aerodynamics which results from contouring the blade
surface to optimize the performance of the blade.
~,

1 ~ 7t~r)75
- - 2
Scientists and engineers are also interested in im-
proving the performance of arrays of airfoils by contour-
ing adjacent airfoil surfaces. Generally the efforts
have fallen into two areas: one, attempting to precisely
define the contours of each airfoil section at almost
every point to optimize the flow relationship between the
airfoil and the working medium gas; the other, generating
airfoils having simple shapes which have better flow
characterietics than conventional shapes. Examples of
botn types of airfoil sections are discussed in Stephens
"Application of Supercritical Airfoil Technology to Com-
pressor Cascades: Compzrison of Theoretical and Experi-
mental Results", AIAA Paper No. 78-1138, presented at the
AIAA Eleventh Fluid and Plasma Dynamics Conference,
Seattle, h'ashington, 1978.
Airfoils having sophisticated shapes such as those
made by the first method are difficult and expensive to
design and very expensive to fabricate. Airfoils made by
~he second method, including double circul2r arc
20 airCo~ ls a;ld multiple circular 2rc airfoils, are rela-
tively simple to design and fabricate but a e not 2S aero-
dynamically efficient as those designed by the first
method. Accordingly, in*erest continues in developing an
airfoil having a shape which is relatively simple to
25 ~ener2te znd yet which exhibits good aerodynamic flow per-
formance in a transonic flow field.
Disclosure of Tr.vention
hccording to the present ir.vention, an airfoil sec-
tion has a cambered meanline, a suction surface and a
press~re surface defined by variables which are ~ func-
tion of the location Gf the .irst covered section.
In accordance with the present invention, an air-
Loil sectior. is fabricated by: establishing a cambered
meanl-ne having â first arc and a second zrc tangentially
intersecting the first z~c at a transition point;
establishing a conical chord line extending between the
leading ed?e and the trziling edge of the meanline;

1 3 7 ~ r~ 7 l3
establishing a thickness distribution about the conical
chord line; superimposing the thickness distribution on
the cambered mean line to form a suction surface and a
pressure surface.
In accordance with a particular embodiment of
the invention, there is provided a rotor blade having
one or more airfoil sections, each airfoil section
being one of a plurality of airfoil sections which are
circumferentially spaced a distance tau ( r) apart about
a rotor axis, each airfo,il section having an inlet metal
angle ~1, a total camber angle et, an alpha chord angle
~ch~ a maximum thickness tmaX, g g ~
ing edge, a tangent line TL passing through the leading
edge tangent to the path of rotation, a front chord
line Bf of length bf, and a conical chord line st f
length bt wherein the values of ~ 1, et, r , bt, the
maximum thickness of the airfoil section tmaX, are
known, the rotor blade having one or more airfoil
section geometries determined by the method steps of:
A. establishing a cambered meanline having
a concave side and a convex side and having a
first arc, a second arc and a transition point
TP between the first arc and the second arc,
the first arc being tangent to-the second arc
at said transition point TP by
Aa, determining an initial value for the
alpha chord angle (~chi) which is equal
to the sum of the inlet metal angle (~1)
and one-half of the total camber angle
(~t)' ( chi ~1 + ~t)'
2 2

1 3 ~J575
-- 4 --
Ab, setting the value of the alpha chord
angle ~ hi (~ h = ~ hi)'
Ac~ determining a distance R from the
tangent line TL to the first covered section
as measured along the conical chord line Bt,
the distance ~ being equal to the distance
tau r multiplied by the quantity the sine
of the angle ninety degrees minus the
alpha chord angle (~ = ~sin(90-~ch),
Ad. determining the normalized distance
Lf s to the first covered section by
dividing the distance ~?by the distance
bt'
Ae, obtaining the ratio of the length
bf of the front chord line Bf to the
length bt of the conical chord line Bt
(bf/bt) and the ratio of the front camber
angle (~f~ to the total camber angle ~t
(~f/~t) as a function of the value Lf
of the normalized distance to the first
covered section,
Af. establishing the location of the first
arc such that the arc passes through the
leading edge using the values known (bt,
~t' ~1) and the value found in step Ae. for
bf and ~f~
Ag, establishing the location of the second
arc such that the arc passes through the
trailing edge using the values known (bt,
~t' ~1) and values fo~nd in step Ae. for
bf, ~f~
Ah, establishing a conical chord line Bt
extending between the leading edge and the
trailing edge,
., .

:-- 1 17~')75
Ai. determining the actual alpha chord
angle ~cha for the cambered meanline,
Aj. determining the difference E between
the actual alpha chord angle ~cha and the
alpha chord angle ~ch used to calculate the
normalized location LfCs by subtracting
~ch from ~cha (E = ~cha-~ch)'
Ak. proceeding to step B if the absolute
value of E is less than the predetermined
value e (/E/<e) and proceeding to step Am
if the absolute value of E is greater than
or equal to the predetermined value
e (IEI~e),
Al. setting the value of the alpha chord
angle ~ch equal to the value ~cha
( ch cha)~
Am. r~peating steps Ac through Aj;
B. establishing a thickness distribution TD
having a line spaced a distance Tzn from the
conical chord line Bt at any point zn, the point
zn being spaced a distance Lan from the leading
edge on the conical chord line Bt, the distance

1 ~7~575
Tzn being measured along a line Zn perpendicular
to the conical chord line st,
C. superimposing the thickness distribution
on the cambered meanline by
Ca, establishing a plurality of points
zn', each point zn' being at the inter-
section of the line Zn and the cambered
meanline,
Cb. establishing a line Z'n perpendicular
to the cambered meanline at each point zn',
Cc, establishing a point zn'` at a distance
Tzn as measured along the line Z'n from
the convex side of the cambered meanline
at each point zn' and a point zn"' at a
distance Tzn as measured along the line Z'n
from the concave side of the cambered mean-
line at each point zn',
Cd, establishing a concave surface passing
through the leading edge and the trailing
edge and through all points zn",
Ce. establishing a convex surface passing
through the leading edge and the trailing
edge and through all points zn"',
In accordance with a further embodiment of the
invention, there is provided a rotor blade having one
or more airfoil sections, each airfoil section being
one of a plurality of airfoil sections which are
circumferentially spaced a distance tau ( r) apart about
a rotor axis each of the airfoil sections having an
inlet metal angle ~1, a total camber angle et, an alpha
chord angle ~ch' a maximum thickness tmaX, a leading
edge, a trailing edge, a tangent line TL passing through
the leading edge tangent to the path of rotation, a
.,, :

5 7 5
-- 7 --
front chord line Bf of length bf, a conical chord line
Bt of length bt wherein the values of ~ t~ ~ ~ bt'
the maximum thickness of the airfoil section tmaX, are
known, the rotor blade having one or more airfoil
section geometries determined by the method steps of:
A. establishing a cambered meanline having a
concave side and a convex side and having a
first arc, a second arc and a transition point
TP between the first arc and the second arc,
the first arc being tangent to the second arc
at said transition point TP by
Aa, determining an initial value for the
alpha chord angle (~chi) which is equal to
the sum of the inlet metal angle (~1) and
one-half of the total camber angle (t)'
( chi ~1 + ~t)~ 2
Ab, setting the value of the alpha chord
angle ~chi (~ch = ~chi)'
Ac, determining a distance~ from the
tangent line TL to the first covered section
as measured along the conical chord line ~t'
the distance ~ being equal to the distance
tau ~ multiplied by the quantity the sine
of the angle ninety degrees minus the alpha
chord angle (~ = ~sin(90-~ch),

; ~
8 _
~ ~7~5'75
Ad, determing the normalized distance LfCs
to the first covered section by dividing
the distance,e by the distance bt,
Ae, obtaining the ratio of the length bf
of the front chord Bf to the length bt
of the conical chord line Bt (bf/bt) and
the ratio of the front camber angle (~f)
to the total camber angle et (ef/~t)
at the value LfCs of the normalized distance
to the first covered section,
Af, establishing the location of the first
arc such that the arc passes through the
leading edge using the values known
(bt' ~t~ ~1) and the value found in step
Ae, for bf and Of~
Ag. establishing the location of the
second arc such that the arc passes through
the trailing edge using the values known
(bt' t' ~1) and values found in step Ae.
for bf, ~f~
Ah. establishing a conical chord line Bt
extending between the leading edge and the
trailing edge,
Ai. determining the actual alpha chord
angle Acha for the cambered meanline,
Aj. determining the difference E between
the actual alpha chord angle ~cha and the
alpha chord angle Ach used to calculate
the normalized location LfCs by subtract-
ing Ach from Acha (E = Acha ch)'
Ak. proceeding to step B if the absolute
value of E is less than the predetermined
value e ~IEl~e) and proceeding to step
Am. if the absolute value of E is greater
than or equal to the predetermined value
e (/E/'e),
, . , .~, .,
'
,
.
. .
- . .
. ; '
". ~ ~ .
`

; ~ 7r~5~5
g
Al setting the value of the alpha chord
angle ~ch equal to the value ~cha
(~ch cha)'
Arn, repeating steps Ac through Aj,
B, establishing a thickness distribution TD
formed of two parts each part being disposed
about the conical chord line B~, each part
having a line spaced Tzn from the conical chord
line Bt at any point zn, the point zn being
spaced a distance Lan from the leading edge
on the conical chord line Bt, the distance
Tzn being measured along a line Zn perpendi-
cular to the conical chord line Bt, the line
of the first part being TDl and the line of
the second part being TD2,
sa, the line of the first part TDl being
established by
Bal, determining the distance loc rnt
along the conical chord line to the
location TMAX of maximum thickness
tmaX by determining the ratio lbC mt
as a function of the value LfCs t
of the normalized distance to the
first covered section,
Ba2, superimposing on the conical
chord line Bt a circle 1'maX having a
center on the conical chord line a
distance equal to loc mt from point
d a s Rtmax equal to one
half of the maximum thickness tmaX
of the airfoil section (RtmaX = tmax),
Ba3, establishing on the conica
chord line Bt a leading edge radius
circle having a radius Rler and a
center on Bt a distance equal to Rler

~ 3 ~57~
-- 10 _
from the leading edge and intersect-
ing the leading edge at a point A, the
radius Rl being equal to a first
constant k multiplied by the maximum
thiCkness tmax (Rler k tmax)'
Ba4, establishing a line Q perpendi-
cular to the conical chord line Bt
at a point which is a distance bf
(Lan = bf) from the leading edge,
sa5. establishing a line F having
a radius of curvature Rf which is
tangent to the leading edge circle at
a point f~, tangent to the circle
TmaX and whieh interseets the line Q
at a point fq,
Ba6. establishing a line P perpendi-
cular to the eonieal ehord line Bt at
a point whieh is a distanee Lan equal
to a second eonstant k2 multiplied by
the length bt of the eonieal ehord
line (Lan = k2 ~ bt) from the leading
edge and whieh interseets the line F
at a point fe,
Ba7. passing the line TDl of the
first part through the points A, fe
and fq sueh that the line of the first
part is tangent to the leading edge
radius cirele at point A, tangent to
the line F at point fe and eoineident
with line F between the points fe and
fq,
Bb. the line of the seeond part TD2 being
established by
,~ .
.

5 7 ~
Bbl, determining the quantity TERG
as a function of the value Lfcs of the
normalized distance to the first
covered section and determining the
radius Rter which is equal to the
quantity TERG multiplied by a third
constant k3 and by tmax
( Rter = TERG ~ 463 tmax)'
Bb2, establishing on the conical
chord line Bt a trailing edge radius
cicle having a radius Rter and a
center on Bt spaced d distance equal
to Rt r from the trailing edge and
intersecting the trailing edge at
a point C,
Bb3, establishing a line G having a
radius of curvature Rg which is
tangent to the trailing edge radius
circle at a point gt and which is
tangent to the line F at the point
fq,
Bb4. passing the line of the second
part TD2 through the points C, gt
an'd fq, such that the line of the
second part is coincident with the
trailing edge radius circle between
the points C and gt and coincident
with the line G between the points
gt and fq,
C, superimposing the thickness distribution on
the cambered meanline by
Ca, establishing a plurality of pOilltS
zn', each point zn' being at the inter-
section of the line Zn and the carnbered
meanlin~,

` ~ ~7~57~
~ 12 -
Cb. establishing a line Z'n perpendicular
to the cambered meanline at each point
zn',
Cc. establishing a point znH at a distance
Tzn as measured along the line Z'n from
the convex side of the cambered meanline
at each point zn' and a point zn"' at a
distance Tzn as measured along the line
Z'n from the concave side of the cambered
meanline at each point zn',
Cd. establishing a concave surface passing
through the leading edge and the trailing
edge and through all points znH,
Ce. establishing a convex surface passing
through the leading edge and the trailing
edge and through all points zn"',
wherein the thickness distribution is stretched
chordwisely on the convex side and compressed on the con-
cave side to form an airfoil section having desirable
20 separation characteristics in a transonic aerodynamic
flow field.

` lg7!~S75
A primary feature of the present invention is a
conical airfoil section having a contoured suction surface
and a contoured pressure surface. Another feature is the
location of the maximum thickness of the airfoil section,
the ratio of the front camber angle ~f to the total camber
angle ~t' the ratio of the length b~ of the front chord
to the length bt of the conical chord line, and the dis-
tance Tzn of the suction surface and the pressure surface
from the cambered meanline.
A principal advantage of the present invention
is the good aerodynamic performance of the airfoil section
in a transonic flow field as compared with circular arc
airfoil sections, Separation of the boundary layer and
the resultant aerodynamic losses are suppressed by con-
trolling the rate of diffusion along the suction surface.
Another advantage is the simple method for generating the
shape of the airfoil as compared with airfoil shapes
yenerated by point by point analysis of the flow field.
The foregoing and other objects, features and
advantages of the present invention will become more
apparent in the light of the following detailed des-
cription of the preferred embodiment thereof as shown
in the accompanying drawing.
Fig. 1 is a developed view of a portion of a
flow directing assembly of a gas turbine engine,
Fig. 2 is a side elevation view of a rotor
blade taken along the line 2-2 as shown in Fig. 1,
Fig. 3 is a sectional view of two adjacent
airfoil sections taken along the line 3-3 of Fig. 2,
Fig. 4 is an enlarged view of the sectional
view of Fig. 3,

~7~575
-14 -
Fis. 5 is a diagra~,atic illustration of the cambered
meanline of the conical zirfoil section of Fig. 4;
Fig. 6 is a graphical representation o' the relatio~-
ship of severzl phvsic21 par~mete~s of the airfoil section
as a func.ion of the n~rmzlized length to the first covered
section ( ~ sin~90-~ch));
Fig. 7 is a diagrammatic view illustrating the sec~nd
step cf forming a ~hic~ness distribution about ~he conical
cnord line Bt;
Fig. 8 is a diagrammatic view corresp~nding to the
~ia~_~matic view of Fig. 7;
Fis. 9 is 2 ciagrammatic view illustrating the step
of ~pply~ng the thickness d~str;bution of Fig. 9 to the
c~mDe_ed meanline o~ Fig. 6;
Fis. 10 is a diagr~mmztic view of the le2ains edge
sesion o -~he thick~ess distribution shown in ~he Fig. 7
and Fis. 8 views.
Best Mode Ior Carsying Out the Invention
~ gas tu_bine engine embodiment o~ a rotary
~2chine ~s illustra~ed in Fig. 1. A portion o 2
10w directing ~ssembly such as a compressor rotor
sss~mhly 10 OI ,he eng~ne is shown. The broken lines
show the embodiment in an unde~eloped view. The solid
l'~es show the e~odiment in the developed view, The
rotor ~ssembly includes z rotor disk 12 having 2~ 2xis
of roLztion R. A plurzlity o' ro~o~ blades 2S re-
presen.ed b~ t:~e roto- blzdes 1~ exLen~ outw2rdlv from
the roLor disk. A llow ?2th 16 .or wor~:ing ~ecium &zses
ex~er.ds be~wee~ ~Lhe adjacent rotor blades. Each b~ade
h~s ~ air~oil 1~ extending ou~ zrdly zc~oss tne wo.king
mediu3 Clow path. Each zir~oil hzs 2 convex su~fzce or
side s~ch zs s~c~ion suIface 2~ znd z c~ncave s~'zAe
o, side such ~s pressu-e su rzce 22.
~.s illustlated in ~ig. 2, the suction surl2ce 20
c~c th~ p-essure su ~2ce 22 of e2ch airfoil 18 c~e

~7~51.)
.
_ 13 --
joined together at a leading edge 24 and a trailing edge
26. An imaginary streamline S in the flow path is adjacent
to each airfoil. An imaginary point A lies on the leading
edge of each airfoil along the streamline S. Point A has
a radius r about the axis R of the engine. Similarly, an
imaginary point B lies on the suction side and an imagi-
nary point C lies on the trailing edge along the stream-
line S. The three points define a section plane S' (3-3).
The plane S' passes through each airfoil and forms a
conical airfoil section 28.
Fig. 3 is a sectional view of two adjacent airfoil
sections 28 taken along the line 3-3 of Fig. 2.
Fig. 4 is an enlarged view of the sectional view of
Fig. 3. The conical chord line Bt is a straight line
connecting point A on the leading edge with point C on the
trailing edge. The conical chord line Bt has a length
bt. A mean camber line such as the cambered meanline MCL
connects the point A on the leading edge and the point C
on the trailing edge. The suction surface 20 and pressure
surface 22 are each spaced a distance Tzn from the cam-
bered meanline along a line Z'n perpendicular to the cam-
bered meanline MCL.
A forward tangent line TL, tangent to the path of
rotation of point A about the axis of rotation R, pro-
vides a reference axis ~y-axis) for measuring angles and
distances. A rear tangent line TLR is parallel to the
tangent line TL and passes through point C. A plane pass-
ing through the axis of rotation R intersects the plane S
at a second reference line, the x-axis. Tau (T) is the
distance between airfoil sections 28 measured along TL.
An alpha chord angle ach is the angle between the tangent
line TL and the conical chord line Bt.
An imaginary point FCS is the location of the first
covered section. A distance Qis the distance from point
FCS to point A measured along the conical chord line Bt.
The distance Qis equal to the distance tau ~ multiplied by
the quantity the sine of the angle ninety degrees minus

1 ~73575
- 16 _
the alpha chord angle or ~ = ~sin(90-~ch). A normalized
distance LfCs to the first covered section is the distance
divided by the distance bt (length of the conical chord
line Bt) (Lfcs bt)
The airfoil has a maximum thickness tmaX. The
location TMAX of maximum thic'~ness is on the cambered mean-
line MCL. A circle TmaX having a radius max is tangent
to the suction surface 20 and the pressure surface 22, The
length loc mt to the location of maximum thickness is
measured along the conical chord line Bt.
The working medium gas flowing along the working
medium flow path 16 approaches the airfoil section 28 at an
angle ~1 to the tangent line TL. The cambered mean line
MCL has a tangent line TMcF at the leading (front) edge.
The angle between the tangent line TMCF and the tangent line
TL is the inlet metal angle ~1. Thus, the tangent line TMCF
intersects the tangent line TL at an inlet metal angle ~
The difference between the angle ~1 and the angle ~1 is the
incidence angle i. As shown in Fig. 4 the incidence angle i
is negative.
The working medium gas leaves the airfoil section
at an angle ~2 to the rear tangent line TLR. The cambered
meanline has a tangent line TMCR at the trailing (rear)
edge. The angle between the tangent line TMCR and the rear
tangent line TLR is the exit metal angle ~2. Thus, the
tangent line TMCR intersects the tangent line TL at an
exit metal angle ~2*. The difference between the angle ~2
and the angle ~2 is the deviation angle d.
As shown in Fig. 5, a total camber angle ~t is
the angle between the tangent line TMCF at the leading edge
and the tangent line TMCR at the trailing edge. The total
camber angle t is the measure of the curve of the ca1nbered
meanline and the airfoil section.
The cambered meanline MCL is a double circular
arc having two circular arcs such as a front arc FA and a

~ :~ 7~r~7~
rear arc RA. m e front arc FA has a radius of curvature
RFA. The rear arc RA has a radius of curvature RRA. The
front arc FA is tangent to the rear arc at a point of
intersection. This point of intersection is the transition
point TP of the airfoil section. A tangent line TFC is
tangent to both arcs at the transition point. A front
camber angle ef is the angle between the tangent line TFC
and the tangent line TMCF. A front camber angle ~f is a
measure of the curve of the front arc FA. A front conical
chord line Bf extends between the point A on the leading
edge and the transition point TP. The front chord line
has a length bf.
Fig. 6 is a graphical representation of the
relationship of several physical parameters which describe
the airfoil section as a function of the normalized length
LfCS to the first covered section (LfCS ='~). The norm-
alized length LfCs is a function of both the gap Y to chord
bt ratio ( r ) and the alpha chord angle ~ch More parti-
cularly LfCts is equal to the distance ~ from the tangent
line TL to the first covered section as measured along
the conical chord line Bt, the distance ~ being equal to
the distance rmultiplied by the quantity the sine of the
angle 90 minus the alpha chord angle divided by the
quantity bt, the length of the conical chord line Bt.
The relationship i8 expressed mathematically
~ = r~sin(90-~ h). The equations approximately describ-
bt bt c
ing this relationship are: ,
~* 6`*
et LfCs, O<Lfcs ~.77; ~* - .27, .77 ~ Lf cl 0;
loc mt = . 367-.087 Lfcs, ~Lfcs ~-77; bt ' fcs
bf b
_ = . 61--. 26 LfCs ~ O ~Lfcg ~- 77; bt
LfCs, O<Lfcs ~.77, TERG = .425, .77~L < 1 0
.. , ,.............................................. ~

r)7 5
_ 18 _
Thus, ~rom Fig. 6 which embodies these equations, the
ratio of the front camber angle ~f to the total camber
angle ~t is related to both the alpha chord angle ~ch and
the gap to chord ratio ~~ by the curve ef divided by et.
t
Similarly, the ratio between the length loc mt to the
location of maximum thickness and the length bt of the
conical chord line Bt is related to both the alpha chord
angle ~ch and the gap to chord ratio br by the curve
loc mt/bt. The ratio of the length bf of the front chord
Bf to the length bt of the conical chord line Bt is related
to both the alpha chord angle ~ch and the gap to chord ratio
b by curve f . Similarly, the relationship for the
dimensionless quantity TERG is related to the alpha chord
angle ~ch and the gap to chord ratio br by the curve TERG.
The quantity TERG is used in determining the distances Tzn.
The steps of the method for forming the airfoil
section 28 are summarized in this paragraph as steps A, B,
C and D, These steps are set forth in more detail in the
following paragraphs. The method for forming the airfoil
section 28 begins with step A (Fig. 5), establishing the
cambered meanline MCL such that the meanline~has a first
arc, such as the front arc FA, and a second arc, such as
the rear arc RA, The first arc and the second arc are
tangent to each other at the transition point TP. The
front arc has a leading end such as the leading edge 24
and the rear arc has a trailing end such as the trailing
edge 26. Step A includes establishing a conical chord
line Bt extending between the leading end and the trailing
end of the cambered meanline MCL, The second step is
step B (Fig, 7), establi~hing a thicXness distribution
TD about the conical chord line Bt. The third step is
step C (Fig, 9) superimposing the thickness distribution
on the cambered meanline, Imposing a thickness distri-
,
.

3 ~ 7 ~
:
- 18a-
bution TD generated about the eonieal ehord line on a
eurved line eauses the thiekness distribution to streteh
ehordwisely on the convex or suction side and to com-
press chordwisely on the concave or pressure side. The
resulting airfoil section has a desirable separation
characteristie in a transonie aerodynamie flow field,
The fourth step is step D. In step D, the airfoil
seetion is completed by forming an airfoil
, , ,
' '

-- 19 --
section having the desired contours. These steps are
explored in more detail below.
Preliminary design based on aerodynamic and
structural considerations establishes the following
values: the length of the conical chord line Bt; the
magnitude of the inlet metal angle ~1; the magnitude of
the total camber angle ~1, the gap distance between
adjacent circumferentially spaced airfoil sections tau r;
and the maximum thickness of the airfoil section tmaX.
Referring to Fig. 4 and Fig. 5, the first step is step:
A. establishing a cambered meanline having
a concave side and a convex side and having
a first arc, such as the front arc FA, a
second arc, such as the rear arc RA, and
a transition point TP between the first
arc and the second arc, the first arc
being tangent to the second arc at said
transition point TP by
Aa. determining an initial value
for the alpha chord angle (A hi)
which is egual to the sum of the
inlet metal angle (~) and one-half
of the total camber angle (t)'
(~chi ~1 2
Ab. setting the value of the alpha
chord angle ~chi (~ch ~chi)'
Ac, determining a distance ~ from
the tangent line TL to the first covered
section as measured along the conical chord
line Bt, the dis~ance e being egual to the
distance tau r multiplied by the ~uantity
the sine of the angle ninety degrees minus
the alpha chord angle (~ = r~sin(90-~ch)),
,
.

1 ~ 7 ~
- l9a -
Ad, determining the normalized
distance LfCs to the first covered
section by dividing the distance
by the distance bt,
Ae, obtaining the ratio of the
length b~ of the front chord line
Bf to the length bt of the
j

~7~575
-20--
conical chord line Bt (bf/bt) and the ratio
of the front camber angle (~f) to the total
camber angle ~t (~f/~t) from Fig. 6 at the
value LfCs f the normalized distance to the
first covered section,
Af. establishing the location of the first arc
such that the arc passes through the leading
edge using the values known (bt, ~t~ ~1) and
the value found in step Ae for bf and ~f~
Ag. establishing the location of the second
arc such that the arc passes through the trail-
ing edge using the values known (bt, ~t~ ~1) and
values found in step Ae for bf, ~f~
Ah. establishing the conical chord line Bt
extending between the leading edge and the
trailing edge,
Ai. deter~ining the actual alpha chord angle
acha for the cambered meanline with respect
to the forward tangent line TL,
Aj. determining the difference E between the
actual alpha chord angle aC~la and the alpha
chord angle ach used to calculate the normalized
location LfCs by substracting ~ch from acha
(E = cha~ach)~
Ak. proceeding to step B if the absolute value
of E is less than a predetermined value e
(IEl<e) and proceeding to step Am if the
absolute value of E is greater than or equal to
the predetermined value e ( ¦E I >e),
Al. setting the value of the alpha chord angle
ach equal to the value ~cha (~ch ~cha),

t~7~r)7
-21-
Am. repeating steps Ac through Aj.
The predetermined value e is selected such th t any
variation in the quantities TERG, f, loc mt and ~f
obtained from Fig. 6 is less than + .02. t bt ~
Fig. 7 illustrates the second step of forming a
thickness distribution TD about the conical chord line Bt.
The second step is:
B. establishing a thickness distribution TD formed
of two parts, each part being disposed about the
conical chord line Bt, each part having a line
spaced a distance Tzn from the conical chord line
Bt at any point zn, the point zn being spaced a
distance Lan from the leading edge on the conical
chord line Bt, the distance Tzn being measured along
a line Zn perpendicular to the conical chord line
Bt, the line of the first part being TDl and the
line of the second part being TD2,
Ba. the line of the first part TDl being
established by
Bal. determining the distance loc mt along
the conical chord line to the location
TMAX of maximum thickness tmaX by deter-
mining the ratio lb mt from Fig. 6 at the
value LfCs of the nortmalized distance to
the first covered section,
Ba2. superimposing on the conical chord
line Bt a circle TmaX having a center on
the conical chord line a distance equal to
loc mt from point A and a radius RtmaX
equal to one-half of the maximum thickness
tMaX of the airfoil secti.on (RtmaX = m2aX).

I 1 7~575
-22-
Ba3. establishing on the conical chord
line Bt a leading edge radius circle
having a radius Rler and a center on Bt
a distance equal to Rler from the leading
edge and intersectin~ the leading edge at
a point A, the radius Rler being equal
to the quantity eighteen hundred and
fifty-two ten thousandths ~.1852) multi-
plied by the maximum thickness tmaX
(Rler = .1852-tmaX)~
Ba4. establishin~ a line Q perpendicular
to the conical chord line Bt at a point
which is a distance bf (Lan = bf) from the
leading edge,
Ba5. establishing a line F having a
radius of curvature Rf which is tangent to
the leading edge circle at a point fQ,
tangent to the circle ~x and which
intersects the line Q at a point fq,
Ba6. establishing a line P perpendicular
to the conical chord line Bt at a point
which is a distance Lan equal to the
quantity thirty-five thousandths multiplied
by the length bt of the conical chord line
(Lan = .035bt) from the leading edge and
which intersects the line F at a point fe,
Ba7. passing the line TDl of the first
part through the points A, fe and fq such
that the line of the first part is tangent
to the leading edge radius circle at
point A, tangent to the line F at point fe
and coincident with line F between the
points fe and fq,

57
-23-
Bb. the line of the second part TD2 being
established by
Bbl. determining the quantity TERG from
Fig. 6 at the value LfCs of the normali~ed
distance to the first covered section and
determining the radius Rter which is equal
to the quantity TERG multiplied by four
hundred and sixty-three thousandths (.463)
and by tmax (Rter = TERG- . 463- tmaX),
Bb2. establishing on the conical chord
line Bt a trailing edge radius circle
having a radius Rter and a center on Bt
spaced a distance equal to Rter from the
trailing edge and intersecting the trailing
edge at the point C,
Bb3. establishing a line G having a
radius of curvature Rg which is tangent to
the trailing edge radius circle at a point
gt and which is tangent to the line F at
the point fq,
Bb4. passing the line of the second part
TD2 through the points C, gt and fq, such
that the line of the second part is
coincident with the trailing edge radius
circle between the points C and gt and
coincident with the line G between the
points gt and fq,
Fig. 8 shows the thickness distribution TD generated
by the preceding step B. The thickness distribution is
disposed about the conical chord line Bt of len~th bt.

~ ~ 7~575
-2~
At point A on the leadin~ edge, the thickness Tzn is equal
to zero (Tzn c Tza = 0). At point C on the trailing edge,
the thickness is equal to zero (Tzn = Tzc = 0). At point
Zl (n=l) a distance Lal from the leading edge A as
measured along the conical chord line Bt (Lan = Lal), the
thickness is equal to Tzl. The distance Tzl is measured
along a line Zl perpendicular to Bt. Similarly, the
thickness of the thickness distribution is equal to Tz2
at point Z2 a distance La2 from the leading edge and Tz3
at point Z3 a distance La3 from the leading edge.
Fig. 9 illustrates the third step of applying (super-
imposing) the thickness distribution on the cambered
meanline to form a convex surface 20 (suction surface) and
a concave surface 22 (pressure surface) of the airfoil
section. The third step is step:
C. superimposing the thickness distribution on the
cambered meanline by
Ca. establishing a plurality of points zn',
each point zn' being at the intersection of the
line Zn and the cambered meanline,
Cb. establishing a line Zn' perpendicular
to the cambered meanline at each point zn',
Cc. establishing a point zn" at a distance Tzn
as measured along the line Zn'from the convex
side of the cambered meanline at each point zn'
and a point zn"' at a dis~ce Tzn as measured
along the line Zn'from the concave side of the
cambered meanline at each point zn',
Cd. establishing a concave surface passing
through the leading edge and the trailing edge
and through all points zn",

~)r)75
-25-
Ce. establishing a convex surface passing
through the leading edge and the trailing edge
and through all points zn"'.
As shown in Fig. 9, the distance between points Zl"
and Z2~ is larger than the distance be~een points Zl and
Z2 on the conical chord line Bt. Thus, the thickness
distribution TD about the conical chord line Bt iss~et~ed
chordwisely on t'ne convex side. The distance between the
points Zl~ and Zi~ is smaller than the distance between
the points Zl and Z2 on the conical chord line Bt. Thus,
the thickness distribution TD about the conical chord
line Bt is compressed chordwisely on the concave side.
An airfoil having a desired separation characteristic
in a transonic aerodynamic flow fie~d results from forming
an airfoil section having a cambered meanline, a convex
surface and a concave surface as established in steps A,
B, C and combining these sections to form an airfoil. The
airfoil is formed in any suitable manner, such as by
casting or casting and machining. The conical airfoil
section 28 as shown in Fig. 4 nas:
a convex surface 20,
a concave surface 22 joined to t~e convex
surface at the leading edge 24 and the trailing edge
26,
wherein the ratio of the front ca~ber angle ~*f
to the total camber angle ~t is related to both the
alpha chor* angle ~ch and the gap to chord ratio b~
by curve ~ of Fig. 6,
wherein the ratio of the length bf of the chord
Bf to the length bt of the conical chord Bt is
related to both the alpha chord angle ~ch and the
gap to chord ratio b- by curve bb_ of Fig. 6,
t t
wherein the ratio between the length loc mt to
the location of ~aximum thickness and the length bt
of the conical chord Bt is related to both the alpha

1! 3 7~57~
-26-
chord angle ~ch and the gap to chord ratio b~ by curve
loc t Of Fig. 6,
wherein the concave surface of the airfoil sec-
tion and the convex surface of the airfoil section
are each spaced a distance Tzn from any point zn per-
pendicular to the cambered meanline, and
~herein the distance T~n is defined by a thick-
ness distribution TD formed of two parts generated
abcut the conical chord line Bt~ each part at any
point zn'having a line spaced the distance Tzn from
the conical chord line Bt as measured along a line
Zn perpendicular to the conical chord line Bt passing
through the point zn' and a point zn, the point zn b~
spaced a distance Lan from a point A on the leading
edge along the conical chord line ~t~ the line of the
first part being TDl and the line of the second part
being TD2 suc~ that
A. the line TDl of the first part
Al. intersects the leading edge at the
point A,
A2. is tangent at the point ~ to a circle
passing through the point A the circle
h2ving a center on the conical chord line
Bt, and a radius Rler, the radius Rler being
2~ equal to the quzntity eighteen hundred and
fifty-two thousandths (.1852) multiplied by
the maximum thickness tmaX of the airfoil
(Rler = 1852'tmax)~
A3. is tangent to a circle having a center
at the location of maximum thickness TMAX
on Bt a distance loc mt fro~ the point A
(Lan = loc mt) and having a radius RtmaX
eaual to one-half of tne mzximum thickness
tm2X of the airfoil section (RtmaX = ~
., ~

5~t)
-27-
A4. is coincident wi.th a line F at a point
fe, the line F being tangent to the circle
having a radius Rler at a point ~, being
tangent to the circle TmaX and having a
radius of curvature Rf, the p~int fe being
spaced from point A as measured along
the conical chord line Bt a distance equal
to the quantity thirty-five thousandths
multiplied by the distance
bt (Lan = La~ = .035bt),
A5. terminates at a point fq, the point fq
being the point of intersection between the
line of the first part TDl and a line Q,
the line Q being perpendicular to the
conical chord line Bt at a point which is a
distance bf (Lan = bf) from the leading
edge, and
A6. has a radius of curvature Rf between
the point fe and the point fq; and
B. the line TD2 of the second part
Bl. is tangent to the line of the first
part at the point fq,
B2. extends from the point fq having a
radius of curvature Rg,
B3. i,s tangent at a point gt to a circle
passing through a point C on the trailing
edge the circle having a center on the
conical chord line Bt and a radius Rter,
the radius Rter being equal to the quantity
TERG multiplied by four hundred and sixty-

~7~57~
three thousandths and multiplied by the
maximum thickness of the airfoil tmaX
(Rter = TERG-463-tmax)'
B4. is coincident with the circle having
the radius Rter between the point gt and the
point C.
Lines TDl within the purview of this invention are
characterized by: coincidence with the line F between
the points fe and fq; and, tangency between the points
fe and A to the line F and to the circle having a radius
Rler. An example of such a line is the broken line TDl
shown in Fig. 10. This line is coincident with the line
F between fQ and f~ and coincident between points fQ and
A with the circle Rler. Another example of such a line is
a line having a linear portion and curved portions at
regions near the point fl and the point A. A third
example is shown by the solid line in Fig. lO. The solid
line TDl is an elliptical line extending between the
points A and fe. The method for establishing the first
part TDl of the thickness distribution for the elliptical
line includes the steps of:
Ba8. establishing an elliptical line ~ ~hich
is tangent to the line F at the point fe and
tangent to the leading edge radius circle
at point A,
Ba9. passing the line of the first part
through the point fe such that the line of the
first part is coincident with the elliptical
line E between the point A and the point fe.
Accordingly, the line TDl of the first part is
coincident with an elliptical line E. The elliptical line

~ 3 ~57~
.
-29-
is tangent at point A to the circle having a radius Rler.
The elliptical line has a radius of curvature equal to Rf
at the point fe and extends between the point A and the
point fe. Such an elliptical line minimizes the dis-
continuity in curvature at the point of tangential junc-
ture with the line F as compared with the discontinuity
in curvature at the point of tangential juncture between
a circle and the line F.
The airfoil section which results from the applica-
tion of this method will perform better in a transonicaerodynamic flow field than a corresponding circular arc
airfoil for any given application. This airfoil section
is intended for a specific range of Mach numbers from
approximately seven tenths M to nine tenths M (.7M-.9M).
The airfoil section obtains its superior behavior from
the contour of the suction surface. The contour of the
suction surface affects diffusion of the working medium
flow along the suction surface of a compressor stage in
such a way that there is an equal risk of separating the
boundary layer at every point chordwisely. Such a dis-
tribution of diffusion avoids a shock wave and the re-
sultant reco~npression. Thus, the airfoil avoids the
losses occurring with the shock wave and the losses
associated with separating the flow.
Al~hough airfoils desi~ned to the above criteria have particu-
lar utility in transonic flow fields, such airfoils also have
utility in subsonic flcw fields and are within the scope of the
teaching contained herein.
Although the invention has been shown and described with
respect to preferred e~bodiments thereof, it should be understood
by those skilled in the art that various ~ es and cmissions in
the form and detail thereof may be made therein withDut departing
from the spirit and the scope of the invention.
This application is a division of application Serial
35 No. 386,108, filed September 17, 1981.