CA1170575A - Airfoil shape for arrays of airfoils - Google Patents

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.

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 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 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 .,, :

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 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 , , , ' ' 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 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 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 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, 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 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~

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) 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~
.

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.

Claims (2)

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:

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 , (.alpha.chi = .beta.? + , 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, 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 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'''.

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 (.alpha.chi = .beta.? + , 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. 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), 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 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 = ), 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 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 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, 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.