a computer program for the specification of axial ...been developed for use in the design of axial...
TRANSCRIPT
l'I
AD-756 879
A Computer Program for the
Specification of Axial
Compressor Airfoils
Aerospace Research Laboratories
DECEMBER 1972
Dist-ibuted By:
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ARL 72-0171
DEcEMMER 1972
Aerospace Research Laborataries
A COMPUTER PROGRAM FOR THE SPECIFICATIONOF AXIAL COMPRESSOR AIRFOILS
GEORGE R. FROST. IS? LT. USAF
RICHARDW . HEARSEY
ARTHUR . WE'I-ERSrROM
FL*UIDDYNA.MICS FA. LITiES RESEARCH LABORATORY
N L TE iINICALPROJECT 7065 TNF: AT N P V 5E ;CE
Approved for public release; distrbution unlId.
AIR FORCE SYSTMS COMMAND
United States Air Force
[4 4
NOTICES
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AIR FORCE/56780/3 January 1973-200
UNCT,AS5T'PTF)se'urit v rlassifieation
DOCUMENT CONTROL DATA . R & DISecuriey classiticator of tie, body of abstret and Indexing anirotallon mul be entered when the overall report Is clostilled)
I CRIGINATING ACTIVITY (Corporate author) 1t.. REPORT SECURITY CLASSIFICATIONFluid Dynamics Facilities Research Laboratory URCT.ASSTITPDAerospace Research Laboratories Ib. GOUP
Wright-Patterson AFB, Ohio 45433 1
3 RIEPORT TITLE
A COMPUTER PROGRAM FOR THE SPECIFICATION OF AXIAL COMPRESSOR AIRFOILS
4 DESCRIPTIVE NOTES (7'pe of report and Inclusive dates)
Scientific FinalS. AU THORIS) (First naone, middle initial. last narae)
George R. Frost, 1st Lt USAFRichard M. HearseyArthur J. Wennerstrom
8 REPORT DATE 7. TOTAL NO PAGCT 7b* NO. O' REFS
December 1972 1148a. r ONTRACT OR GRANT NO 90. ORIGINATOR'S REPORT NUMOCRIS)
In-house Researchb. PROJECT NO. ARL 72-01717065-0 -09c. DOD Element: 61102F 9b. OTHER REPORT NOIS) (Any other numbers that may be assigned
thl report)
d. DOD Subelement: 68130710 OISTRPOUTION STATEMENT
This document has been approved for public release and sale;its distribution is unlimited.
11 SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY
TECH OTHER Aerospace Research Laboratories (LF)Wright-Patterson AFB, Ohio
13. AOSTRACT
This report is a revision and extension of ARL 70-0046. Itdescribes the analysis in, and the use of, a computer program which hasbeen developed for use in the design of axial compressor airfoilssuitable for operation at high subsonic and supersonic Mach numbers.Four rather versatile camber line shapes and two thickness distributionsare mathematically derived. These camber lines provide the capabilityof defining a wide variety of blades, from those of continuouslypositive camber to the so-called "S-blades", including many of theintermediate possibilities. A method is presented whereby theairfoils are specified on arbitrary axisymmetric streamsurfacesand then accurately redetermined in Cartesian coordinates on planesnormal to the stacking axis. The program determines the coordinates,calculates section properties, and provides precision plots of thespecified blade sections and the resulting blade projections on theabove-mentioned planes.
DD Nov 1473 UNCLASTFTEDSecurity Classification
UNCLASSIFIEDSecurity Classification
14. L INK A | LINK I LINK CKEY WOftOS ....... 4 - -
SROL WT ROLE WT ROLE WT
axial compressor
compressor blade
airfoil geometry
turbine engines
turbomachinery
gas turbines
*U.S.Government Printing Office: 1972 - 759489/265 UNCLASSIFIEDSecurity Classification
. .
ARL 72-0171
A COMPUTER PROGRAMFOR THE SPECIFICATIONOF AXIAL COMPRESSOR AIRFOILS
GEORGE R. FROST, IST LT, USAF
RICHARD M. HEARSEY
ARTHUR J. WENNERSTROM
FLUID DYNAMICS FACILITIES RESEARCH LABORATORY
DECEMBER 1972
PROJECT 7065
Approved for public release; distribution unlimited.
AEROSPACE RESEARCH LABORATORIESAIR FORCE SYSTEMS COMMAND
UNITED STATES AIR FORCEWRIGHT-PATrERSON AIR FORCE BASE, OHIO
IC-
r,
FOREWORD
This report was prepared by Ist Lt. George R. Frost andDr. Arthur J. Wennerstrom off the Fluid Dynamics FacilitiesResearch Laboratory, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio, and by Mr. Richard M. Hearseyof the University of Dayton Research Institute.
The report presents resultts from a portion of the effortof the Fluid Machinery Research Group, supervised by Dr. ArthurJ. Wennerstrom and was conducted under Work Unit 09 of Project7065, "Aerospace Simulation Techniques Research" under theoverall direction of Mr. Elmer G. Johnson.
ii
ABSTRACT
This report is a revision and extension of ARL 70-0046.It describes the analysis in, and the use of, a computerprogram which has been developed for use in the design of axialcompressor airfoils suitable for operation at high subsonic andsupersonic Mach numbers. Four rather versatile camber lineshapes and two thickness distributions are mathematicallyderived. These camber lines provide the capability of defininga wide variety of blades, from those of continuously positivecamber to the so-called "S-blades", including many of theintermediate possibilities. A method is presented whereby theairfoils are specified on arbitrary axisymmetric streamsurfacesand then accurately redetermined in Cartesian coordinates onplanes normal to the stacking axis. The program determines thecoordinates. calculates section properties, and providesprecision plots of the specified blade sections and the result-ing blade projections on the above-mentioned planes.
iii
TABLE OF CONTENTS
SECTION PAGE
I INTRODUCTION ......... .................. 1
II THE SECTION CAMBER LINE ..... ............ 4
1. Polynomial Camber Line .. ............. 5
2. Exponential Camber Line .... .......... 7
3. Circular-Arc Camber Line ..... .......... 10
4. Multiple-Circular-Arc Camber Line ..... .. 13
III THE SECTION THICKNESS DISTRIBUTION ... ....... 15
1. Standard Thickness Distribution . ...... .. 15
2. Double-Circular-Arc Thickness Distribution . 18
IV CARTESIAN COORDINATES FOR THE BLADE . ...... .. 22
V SECTION PROPERTIES ..... .............. ... 27
1. Streamsurface Sections ... .......... ... 27
2. Manufacturing Sections ... ........... ... 30
VI BLADE CHARACTERISTICS .... ............. .. 33
1. Blade Volume Calculation .. .......... ... 33
2. Calculations for Aerodynamic Analysis . . . 34
VII USE OF THE COMPUTER PROGRAM ... .......... .. 35
1. Definition of Input Data Items ........ . 35
2. Input Data Format .... ............. .. 39
3. Output Data ..... ................ .. 40
VIII EXAMPLES OF USE OF THE PROGRAM ........... ... 42
1. Introduction ..... ................ .. 42
2. Polynomial Camber Line .. ........... ... 42
a. Input Data .... ............... ... 42
Preceding page blank
SECTION PAGE
b. Output Data ...... ............... 46
3. Exponential Camber Line .... .......... ... 62
a. Input Data ...... ............. ... 62
b. Output Data ..... ............... ... 66
4. Double-Circular-Arc Blade (Circular-ArcCamberline) ...... ................. ... 66
a. Input Data ... .. ............... ... 66
b. Output Data ...... .............. ... 69
5. Multiple-Circular-Arc Camber Line .. ...... 74
a. Input Data ...... ............... 74
b. Output Data ... .. .............. . 78
IX COMPUTER PROGRAM DETAILS ... ............ ... 79
I. Implementation of the Computer Program . . . 79
2. Deck Setup for CDC 6000 Series Computer . . . 80
3. FORTRAN Program Listings .. .......... ... 81
4. Program Logic ...... ................ ... 124
ILLUSTRATIONS ....... .................. ... 132
REFERENCES ....... ................... ... 158
vi
ILLUSTRATIONS
FIGURE PAGE
1 Variation of Normalized Second Derivative ofPolynomial Camber Line with Parameter "P" ..... 132
2 Variation of Polynomial Camber Line withParameter "P" . . . . . . . . . . . . . . . . . .. . . . . . . . . 133
3 Variation of Polynomial Camber Line withParameter "P" ....... ................... .. 1314
14 Variation of Polynomial Camber Line withParameter "Q" .......... .................. ..135
5 Variation of Polynomial Camber Line with ALPHA2 136
6 Variation of Normalized Second Derivative ofExponential Camber Line with Parameters"P" and "Q"'' ........ .................... .. 137
7 Variation of Exponential Camber Line withParameters "P" and "Q . . . . . . . .. . . . . .. . . . . . .. 138
8 Variation of Exponential Camber Line withParameters "P", and "Q". ..... .............. .. 139
9 Variation of Exponential Camber Line withParameter "S ......... ................... 140
10 Variation of Exponential Camber Line withParameter "S" .......... ................... 141
11 Some Camber Lines Available under Multiple-Circular-Arc Camber Line Option ... .......... .142
12 Variation of Multiple-Circular-Arc Camber Linewith Parameter "S" ....... ................ .1143
13 Variation of Multiple-Circular-Arc Camber Linewith Parameter "S ....... ................ .. 1144
14 Variation of Thickness Distribution with Pointof Maximum Thickness ..... ............... .1145
15 Cartesian and Streamsurface Coordinates of a
Point ......... ...................... ... 146
16 Calculation of Section Properties .. ......... .. 1147
17 Parameters Describing Blade Sections ... ....... 148
vii
FIGURE PAGE
18 Locations of Streamsurfaces and ManufacturingPlanes for Example Blade Designs ... ........ .1149
19 Example Blade Design: Polynomial Camber Line(Streamsurface Sections) ..... ............. .150
20 Example Blade Design: Polynomial Camber Line(Manufacturing Sections) ...... ............. 151
21 Example Blade Design: Exponential Camber Line(Streamsurface Sections) ..... .......... ... 152
22 Example Blade Design: Eyponential Camber Line(Manufacturing Sections) ..... ............. .. 153
23 Example Blade Design: Double-Circular-ArcBlade (Streamsurface Sections) .... .......... .1511
24 Example Blade Design: Double-Circular-Arc Blade(Manufacturing Sections) ....... ............. 155
25 Example Blade Design: Multiple-Circular-ArcCamber Line (Streamsurface Sections) .. ....... .. 156
26 Example Blade Design: Multiple-Circular-ArcCamber Line (Manufacturing Sections) . ....... ... 157
viii
SECTION I
INTRODUCTION
The Fluid Dynamics Facilities Research Laboratory of theAerospace Research Laboratories (ARL) is engaged in a programto find means of designing axial compressors capable of efficientperformance at Mach numbers and pressure ratios beyond thecurrent state-of-the-art. In conjunction with this effort, itwas determined that general improvements could be made oversome of the blade-design techniques now in popular use for highrelative Mach numbers.
it is desirable when designing a compressor blade to createblade sections upon arbitrary surfaces of revolution thatcorrespond to streamsurfaces indicated by the compressor designcalculations. However, for manufacturing purposes, it isdesirable to have blade surface coordinates for sectionsthrough the blade that are plane and perpendicular to thestacking axis. A method is presented whereby the blade isdesigned by creating sections on the streamsurfaces, and thendetermining coordinates for plane sections through the resultantblade.
Several objectives guide the aerodynamic design of anyairfoil, whether for a compressor or any other purpose. First,and usually foremost, is the desire to keep the rate ofdiffusion as low as possible on the airfoil surfaces. Whilethis objective does not change with Mach number, the shape ofairfoil required to accomplish this can change significantlywith Mach number. At supersonic Mach numbers there is generallylittle camber, and sometimes negative camber, in the leadingedge region of a compressor blade. At these Mach numbers,minimizing diffusion rates means minimizing the suction surfacesupersonic expansion upstream of the inevitable passage shockor even producing a limited amount of supersonic diffusionupstream of the shock location. There are two principal methodsof obtaining supersonic diffusion. When using blades havingzero or positive camber near the leading edge, the diffusionmay be obtained by a convergence of the annulus walls. If thisis impractical, for example, in a high-aspect-ratio supersonicfan rotor, the same result can be accomplished by negativecamber in the leading edge region of the airfoil. Blades whichhave negative camber followed by a region of positive camberhave been called "S-blades".
A second objective is to define the airfoil in such amanner that its shape can be continuously varied to accommodatevarying conditions along the blade span, thus maintainingresonable diffusion rates along each blade element. Forexample, it would be possible to define a type of profile
1
having a very attractive diffusion distribution for a hub section,but which could not be adapted to tip conditions, or vice versa.This would obviously be undesirable. It would generally beimpractical to attempt to blend airfoil sections of more thanone type along the span of a blade. A third objective for adesirable compressor blade is that the parameters determiningits aerodynamic characteristics should be variable over a widerange within which there is little danger of producing shapesthat are mechanically undesirable.
This report is a revision and extension of an earlierreport (Reference 1) which presented two blade profiles whichsatisfy all the above requirements. They are composed of acommon thickness distribution applied to two different camberlines: one a fourth-order polynomial, the other defined by twoexponential functions. The thickness distribution consists oftwo third-order polynomials: one between the leading edge andthe point of maximum thickness, and one from there to thetrailing edge.
In addition to these camber lines, two other camber lineshave been incorporated into the associated computer program.One of these is the familiar circular-arc camber line, uponwhic-h is placed the double-circular-arc thickness distribution.The second camber line may be composed of any pair of circulararcs, either of which may degenerate to a straight line,allowing for a wide range of design flexibility. This lattercamber line, referred to as the multiple-circular-arc camberline in this report, uses the common thickness distribution ofthe original blade profiles and, as a special case, allowsthis thickness distribution to be applied to a simple circular-arc camber line.
The various camber lines and thickness distributions aredescribed in Sections Ii and III.
In order to specify a blade for manufacture it is desirableto have coordinates for plane sections through the blade. Formaximum accuracy, blade profiles should be designed on thearbitrary surfaces of eevolution that typical current designprocedures indicate. The method of satisfying these tworequirements consists first of designing blade profiles onsurfaces of revolution. Then a thin, uniformly-flexible beamis passed (mathematically) through like points on each profileand the intersection of the beam with the "manufacturingplanes" yields the coordinates of the "manufacturing section:s."The original report and computer program presented Equation ( 2 )Reference 1, incorrectly which resulted in an improperdetermination of the Cartesian coordinates of the specifiedstreamsurface blade sections for any non-cylindrical stream-surface. This led to associated errors in the determination ofthe manufacturing sections. (The magnitude of this error
C4
increases with streamsurface slope angle and generally wouldnot result in a blade leading or trailing edge angle error ofmore than about one degree.) The version of the program present-ed in this report incorporates the correct analysis as describedin Section IV.
The fifth section of this report describes calculations todetermine various properties of both the streamsurface bladeprofiles and the manufacturing sections. Several bladecharacteristics required for aerodynamic analysis of a resultingblade row are presented in Section VI.
In addition to the new features and the analysis correctionalready mentioned, the computer program (presented in SectionsVII through IX) has been modified in several other respectssince the original report was prepared. The program nowaccommodates curvilinear computing stations, as compared tothe strictly linear stations previously required. An additionalsection property, the torsional constant, is calculated forthe manufacturing sections to facilitate structural analysis ofthe blade. Finally, a blade may now be stacked also at eitherthe leading or trailing edge, in addition to the previouscaoability of stacking at, or offset from, the -ection centroid.
The seventh section of this report presents a generaluser's guide for the computer program, 3nd Section VIII presentsseveral examples for the use of the program. The actual codingand a synopsis of the essential steps in the program logic,as well as information which may facilitate its implemontationon a computing system, are presented in Section IX.
increases with streansurface slope angle and generally wouldnot result in a blade leading or trailing edre angle error ofmore than about one degree.) The version of the program present-ed in this report incorporates the correct analysis as describedin Section IV.
The fifth section of this report describes calculations todetermine various properties of boh the streamsurface bladeprofiles and the manufacturing sections. Several bladecharacte-istics required for aerodynamic analysis of a resultingblade o are presented in Section VI.
In addi-ion to the net. features and the analysis correctionalready mentioned, the computer program (presented in SectionsVII through IX) has been modified in several other lespectssi,,,ce tie original report was prepared. The program nowaccommodates curvilinear computing stations, as compared tothe strictly linear stations previously required. An additionalsection property, the torsional constant, is calculated forthe manufacturing sections to facilitate structural analysis ofthe blade. Finally, a blade may now be stacked also at eitherthe leading or trailing edge, in addition to the previouscapability of stacking at, or offset fro,., the section centroid.
The seventh section of this report presents a generaluser's guide for the computer program, and Section VIII presentsseveral examples for the use of the program. The actual codingand a synopsis of the essential steps in the program logic,as well as information which may facilitate its implementationon a computing system, are presented in Section IX.
4
i.
Qd
SECTION II
THE SECTION CAMBER LINE
To obtain a camber line varying smoothly in curvature andsatisfying the requirements of a compressor blade, it would beconvenient to describe it mathematically in equation form. Thetype and order of such an equation will in turn be influencedby the boundary conditions which are imposed. The equationmust be of' high enough order to accommodate all of thenecessary boundary conditions. Its order should be no higherthan absolutely necessary, however, to minimize the number ofsingularities which can occur within its useful working range.In the following discussion, we shall consider an equation ofthe form y = f(x) where the variable x is nondimensional andvaries from 0 at the blade leading edge to 1.0 at the bladetrailing edge. Furthermore, instead of defining the x-axis asthe chord line, which is usual, x has been defined as the axialdirection in the blade-element plane. This definition of thex-axis was chosen because it was found to lead to the simplesttreatment of boundary conditions.
Three boundary conditions, or their equivalent, areabsolutely necessary. One condition fixes a point on the linewith respect to the coordinate system, another defines theslope at the leading edge, and the third defines the slope atthe trailing edge. These three alone are sufficient to definea unique circu].ar-arc camber line between the edge points.However, these boundary conditions are insufficient to definea camber line for a blade which will meet the special require-ments of efficient operation at relatively high Mach numbers.
A characteristic usually desired for supersonic compressorblades is very little camber in the leading portion of theblade. Depending upon the overall aerodynamic design, thedesired leading-edge camber may be negative, zero, or slightlypositive. A convenient way of controlling this is to imposea boundary condition on the second derivative of the camberline at the leading edge. To put it in dimensionless form, wespecify the ratio between second derivatives at the leadingedge and the point where the absolute value of the secondderivative io a maximum. A fifth condition was felt advisablefor an efficient compressor blade. Namely, the first fourconditions used alone tend to produce a blade whose curvatureis highest at the trailing edge. This could result in largedeviation angles and correopondinigly high losses. Therefore,a condition wa imposed on the -econd derivative at the trailingedge whereby it. value could be specified a," something lessthan the maximum. Thio boundary condition was phrased inexactly the -ante manner as at th.? leading edge. The resultingfive conditions may be written
a4
At x 0: y =0 (1)
y' = tan al
Y"= P (Y1") max
At x = 1: y' = tan a2 (2)
Y= Q (Y") max
where prime (t) and double prime (") refer to first and secondderivatives respectively.
1. POLYNOMIAL CAMBER LINE
The five stated boundary conditions could be applied to agreat number of equations to produce a camber line. A simplefourth order polynomial was chosen for the blade havingprimarily positive camber because it was convenient, and provedquite satisfactory. Since the second derivative of a fourthorder, polynomial is obviously second order and two of theboundary conditions apply to the second derivative, it wasconvenient to start by phrasing this in the form of a parabola.In standard form this can be written
(x - h)2 = 4a (y" - k) (3)
orl y" - (x - h)2 + k4a
where h is the position on the x-axis where the secondderivative is a maximum (or minimum) and k is the value whichthe second derivative has at this point. Integrating twice,one obtains
Y= 12 (x - h)3 + kx + b (4)
Y 14 (x - h )4 + k x2 + bx + c (5)48a -2
Application of the five boundary conditions to these equationsproduces the following result
At x = 0, y = 0 (6)
h 4
48a
At x = 0, y' = tan a (7)
h3
b = + tan a1
At x = 0, Y" = P (Y")max
h2 (8)
4a (I-P)
At x = 1, y' = tan a2 (9)
a I h 2 + h --a= 4 (tan a -tan a) 3
At x = 1, Y" Q (Y"max (10)
h =
The values of the input parameters at which the equationsdegenerate are easily ascertained by examining the derivedconstants. The values of the two blade angles may be anythingbetween, and less than, ±90 degrees, measured fron. the axialdirection. Note that when the two blade angles are the same,this fourth-order camber line becomes a straight line,regardless of what values P and Q may have. The parameter Qmay be given any value between zero and 1.0. In practice, avalue of 0.5 has proven very satisfactory for most applications.Singularities associated with the parameter P are partiallydependent upon Q. At P = 1.0, a singularity exists independentlyof Q. However, if Q = 1.0, the equations also degenerate atP = -2.0 and if Q = 0, the equations degenerate at P = -3.0.In general, P should be greater than -2.0 and less than 1.0.For most applications, a value of P = 0 is quite satisfactory.Toward the hub of a compressor, it may be desirable to give Psmall positive values. Toward the tip, one might wish to useslightly negative values of P. However, this particularcamber line does not produce an attractive "S-blade," especially
6
if the overall section camber is nearly zero. fleg-ative valuesof P are only useful for delaying the onset of positive curvatureby producing very slight negative cambers at th? leadinr edge.
Figures 1 through 5 illustrate the general characteristicsof this camber line. In Figure 1, the distribution of the secondderivative of the camber line is shown for various values of Pand blade angles character.stic of mid span. Figure 2 showsthe actual camber lines which correspond to the curves shown inFigure 1. Figure 3 is similar to Figure 2 but shows theinfluence of P on a camber line more typical of a compressorhub section. Figure 4 illustrates the influence of Q for acamber line typical of mid span with P = zero. Finally,Figure 5 shows the different camber lines which result fromvarying the blade outlet angle with all other parameters fixedat typical values.
2. EXPONENTIAL CAMBER LINE
The exponential camber line is the result of efforts tofind a mathematical expression capable of defining a satisfactory"S-blade." An "S-blade" should satisfy all of the basiccriteria which led to the preceding polynomial camber line and,in addition, several conditions unique to the "S" configuration.Namely, it should be pcssible to locate the inflection pointanywhere on the camber line, the blade angle at that pointshould be independently specifiable, and the method must workfor overall blade cambers which are positive, zero, or negative.Furthermore, the transitior across the inflection point shouldoccur smoothly but rapidly so that a long straight region isavoided in the middle of the blade. It is also highly desirablethat the equations allow a smooth transition from the "S" config-uration to a more conventional one such as characterized bythe polynomial camber line. To the list of five boundaryconditions presented earlier, one is added:
At x = s, y' = tan a (11)
Numerous equations, including polynomials, were examinedwith respect to the six boundary conditions. It soon becameevident that more satisfactory results could be obtained bydefining the camber line with two equations, one either sideof the inflection point. As with the previous blade, it waseasiest to choose an equation by examining equations for thesecond derivative of the camber line and then integrating twiceto obtain the coordinate equation. The equation selected forboth portions of' the camber line is
a(x-s) (12)y" = b (x -
s) e
7
One set of constants applies from 0 to s and another set appliesfrom s to 1.0. The maximum value of y" occurs where y''' = 0,which is at
x = S (13)a
Note that the constant a is positive ahead of the inflectionpoint s, and is negative after it. The maximum (or minimum)value of the camber line second derivative for each camber linesegment is
y, - b (l'i)Y max ae
Integrating the second-derivative equation twice, one obtains
y= b ea(x-s) ra(x-s) - 1] + c (15)a2
(16)
Sb 0a(x-S) [a(x-s) - 2] + c (x-s) + dY- a3
This equation contains four arbitrary constants, and there aretwo sets to be determined as described above. Because there aredifferent equations describing the two portions of the camberline, a seventh condition is added to the six already stated.At the inflection point, the coordinates of the camber line arethe same on both lines.
Applying the four conditions appropriate for the leadingportion of the camber line produces the following result:
At x = 0, y= 0
di = (als + 2)-- e (17)al
84
At x = 0, y' = tan a
C1 = tan a +(as + al
At x = 0, y" = P (yamax
P = as(las) (18)
At x = s, y' = tan a (19)S
C 1 = tan a +s al
Equating the two expressions for c, yields
b a12 (tan al - tan as ) (20)
1 1 - (ais + l)e-als
Applying the four conditions appropriate for the rearwardportion of the camber line produces the following result:
At x = s, y' = tan as
c = tan a + 2(21)
At x = s, y = y obtained from Equation (16) for the leadingportion of the camber line.
d = 2 2 1 + s (C - c2) + d1
At x = 1, y' = tan a2
tan - b2 ea2(1 - s ) [a2 (1-s) - 1] (22)
2 = aa2
9
Equating the two expressions for c2 yields
a 2 (tan a2 - tan as ) (23)
2 1+ [a 2 (l-s) - 1] ea2(1- s )
At x 1, y" =Q (y"m)max
Q = (s-1)a 2e l+a2 (1-s) (24)
The equations obtained to determine al and a2 (Equations(18) and (24))are implicit, and require iterative solution.In each case, there are two solutions for the constant (assumingP and Q to be less than unity). Because it is desired that themagnitude of the second derivative reach its maximum valuebetween the inflection point and leading or trailing edge, asappropriate, it follous that
a > (25)S
and1 a
a <? s-1
Figures 6 through 10 illustrate the general characteristicof this camber line. Figure 6 shows the distribution of thesecond derivative of the camber line for various values of P andQ. The other parameters that are held constant describe ablade perhaps typical of mid-span conditions for a high-speedmachine. Figure 7 shows the corresponding camber line.Figure 8 shows camber lines for a section having zero overallcamber and hence equal negative and positive cambers. Theeffect of varying the parameters P and Q is shown. Figure 9shows camber lines for a similar section, except that the effectof varying the location of the inflection point S is illustrated.Camber lines for three sections having no negative camber areshown in Figure 10. As the "inflection point" is shiftedrearwards from the leading edge, the camber line changes fromone of continuous positive camber to that of a "J-blade."
3. CIRCULAR-ARC CAMBER LINE
The circular-arc camber line has been considered becauseof the wide application of the double-circular-arc airfoil inturbo-machinery of the more conventional variety and becauseof its potential use, particularly with respect to stators, in
10
high performance compressor stages. Of the original five boundaryconditions of Equations (1) and (2), those involving y" aresuperfluous to the definition of' a unique circular-arc camberline.
The equation of this camber line is of the form
(x - x0 )2 + (y - yo )? = R2 (26)
where (x0 , y0 ) is the center of, and R the radius of, the circleof which the desired arc is a portion.
From the boundary condition at x = 1 and the knowledge thatthe slope of a radial line at any point must equal the negativereciprocal of the slope of the tangent to the arc at that point,one obtains
(I - x0 ) (27)tan a. - -
Y2 - YO
where Y2 is the y coordinate of the arc at x = 1. Solvingfor x0 ,
x0 = (Y2 - Y0 ) tan a2 +1 (28)
At the point (1, y2 ), Equation (26) may be arranged to yield
(Y2 - Yo )2 = R2 - X2 + 2x0 - 1 (29)
From the boundary condition which fixes the leading edgeof the arc at the origin of the coordinate system, and Equation(26), one obtains
= R 2 (30)
Substituting Equations (28) and (30) into Equation (29),
1 2 2 2(31)(y Y)2 =YO + 2 (y2 - YO) tan a 2 + 1
1I
iJ
L!
Expanding the left hand side of Equation (31) and solving
for y0 ,
y22 - 2y2 tan a2 - 1 (32)
Yo 2 (y2 - tan a 2 )
Using this expression for y 0 in Equation (28),
= (y22 - 1) tan a2 + 2y2 (33)
2(y2 - tan a2 )
and substituting Equations (32) and (33) into (30) and solving
for R2 ,
(Y22 + 1)2 sec2 a2 (34)
4(y 2 - tan a2 ) 2
The above equations (Equations (32), (33) and (34)) givethe necessary camber line constants in terms of the as yetundetermined Y2 . The third boundary condition, fixing theslope at the origin, is employed to express y, in terms ofknown quantities. From this boundary condition,
x0 = -Yo tan al (35)
Combining Equations (28) and (33) and solving for Y2,
(36)Y Y (tan a2 - tan a) - 1
2 tan a,
Substituting this expression for Y 2 into Equation (32)
yields, after considerable simplification,
- tan al sec 2 a 2 ± tan a 2 sec al sec a2 (37)YO = tan 2 at -tan 2 a 2
12
I..
The proper sign of the second terp (-) of Equation (37)is determined by the realization that y01 must approachinfinity as a2 approaches a,. Therefore,
- (tan al sec 2 C. + tan al sec al sec a2) (38)Y tan 2 a1 - tan2 a.2
Substitution of Equation (38) into Equation (35) determinesx0 in terms of known quantities. Substitution of Equation (38)into (36) determines y. in terms of the known quantities aland a2 , which, when used in Equation (34), yields anexpression for P2 in terms of a, and a2 " This completes thedetermination of the three constants (x0 , y0, and R2) of thecircular arc camber line.
4. MULTIPLE-CIRCULAR-ARC CAMBER LINE
The multiple-circular-arc camber line incorporates andextends the circular-arc definition scheme as previouslydescribed to include the potential for defining a camber linecomposed of two segments, either or both of which may be eithercircular arcs or straight lines, in principle. The most interest-ing applications of this capability probably involve thecreation of "S-blades," composed of two circular arcs, and"J-blades" composed of a straight-line and circular-arccombination.
The specification of this two-segment camber line requiresthe boundary condition of Equation (11) as well as the threeboundary conditions necessary for the circular-arc camber line.The circular-arc portions are computed in the manner of thesubsection "Circular-Arc Camber Line" using the inflectionangle and the inlet (or outlet) angle. Each segment is scaled,and positioned with respect to the origin (if necessary). Astraight-line segment is created when the angle at the inflectionpoint is identical to the inlet (or outlet) angle.
One other feature of the programming associated with themultiple-circular-arc camber line should be noted: an ordinarycircular-arc camber line will be produced if the inflectionpoint is specified at the blade leading edge. Since the standardthickness distribution (described in Section III) is appliedto the multiple-circular-arc camber line, this feature enablesthe program user to define a blade using the circular-arc camberline and either the standard or double circular-arc thicknessdistribution.
Figure 11 shows various types of camber lines which may bespecified using the multiple-circular-arc camber line option.
13
Figure 12 shows camber lines for similar "S-blade" sections,illustrating the effect of varying the location of theinflection point. Camber lines for three "J-blade" sectionswith different inflection locations are shown in Figure 13.
14
SECTION III
THE SECTION THICKNESS DISTRIBUTION
1. STANDARD THICKNESS DISTRIBUTION
For the same reasons that it was desirable to define thecamber line in equation form, it is also practical to definethe thickness distribution this way. Five features wereparticularly desired for a supersonic compressor airfoil. First,the maximum thickness should be arbitrary. Second, it shouldbe possible to locate the maximum thickness at any point onthe downstream half of the airfoil. Third, there should be nodiscontinuity in curvature, meaning that derivatives throughthe second should be continuous. Fourth, for mechanical reasons,the airfoil should have a thickness distribution which iscontinuously convex; i.e., the second derivative should notchange sign. Fifth, the airfoil should have the slenderestleading edge that is compatible with the first four conditions.
Numerous equations were examined with these objectives inmind. However, no single equation was found which seemed tosatisfy all of the foregoing requirements to a satisfactoryextent. The method finally selected consisted of definingthe thickness distribution with two equations: one from theleading edge to the point of maximum thickness and one fromthere to the trailing edge. At the * oint of juncture, thethickness and the first and second derivatives of thicknesswere equated. In order to prevent a reflex curvature fromoccurring in the thickness distribution near the leading edge,the second derivative of thickness was set equal to zero at theleading edge. Thickness was considered to be distributednormal to a camber line of unit length. Maximum thickness waslocated at a point Z on this axis which, it should be noted, isnot the chord line. The thickness of the leading and trailingedges is independently specified so that it need not be thesame at both edges. At the leading edge, the blade surface iscompleted with a circular arc. At the trailing edge, the bladewas truncated by connecting the two end points with a straightline.
The boundary conditions chosen for the two segments ofthe thickness distribution may be summarized as follows.
15
For the Leading Portion:
At x 0, y y (39)=t 0
At x = Z, y = T/2
Y' = 0
For the Trailing Portion:
At x = Z, y = T/2 (40)
y 0
It (leading portion)
At x = 1.0, y :y
Note that y represents the half-thickness of the blade elementand T is the total value of the maximum thickness. Both arescaled in proportion to a camber line length of unity. Theparameter Z denotes the position where maximum thickness occurson the camber line. For the leading section, a simple third-order polynomial was used in the form
y = ax3 + bx2 + cx + d (41)
y' = 3ax 2 + 2bx + c
y" = 6ax + 2b
Applying the boundary conditions for this portion produces thefollowing result valid from 0 to Z:
T/2 - y (42)a 0
2Z 3
b 0
3(T/2 - yo) (43)2Z
d =Y (44)
16
Also, at x = Z (45)
= - 3(T/2 - yo)y = Z2
which is used as a boundary condition for the trailing portionof the blade. A similar third-order equation was used todefine the trailinp portion of the thickness distribution. Thiswao arranged
y = e(x-Z) 3 + f(x-Z)2 + g(x-Z) + h (46)
y' = 3e(x-Z)2 + 2f(x-Z) + g
y = 6e(x-Z) + 2f
Applying the boundary conditions relating to the trailing portionleads to the following result valid from x = z to 1.0.
3(T/2-yo) (T/2-yl) (47)
2Z2(I-Z) (l-Z) 3
3(T/2-yo) (48)2Z2
g 0 (49)
h = T/2
As the equations for the two portions of the thicknessdistribution are cubics, potentially there is the possibilityof introducing inflection points into the thickness distribution.One boundary condition applied to the equation describing theforward portion of the thickness was that the second derivativebe zero at the leading edge. Hence, because a cubic has onlyone point of inflection, the thickness distribution forward ofthe point of maximum thickness cannot contain an inflection.Examination of the equation for the second portion of thethickness distribution yields the result that there is a minimumvalue of Z (the point of maximum thickness) such that the in-flection point of this equation is not on the blade surface.This is given by
17
l i (50)
min 1 + 1-2y/T
1-2yo/T
Thus if the leading and trailing edge thicknesses are equal,Z O n is 0.5, and will increase as the ratio of the trailinge ge to leading edge thicknesses increases.
Figure 1I shows thickness distributions for three valuesof Z, with the leading and trailing edge thicknesses equal ineach case.
2. DOUBLE-CIRCULAR-ARC THICKNESS DISTRIBUTION
The determination of the double-circular-arc thicknessdistribution is essentially a geometrical exercise involvingthe translation and rotation of coordinate systems and the useof polar coordinates for the sake of simplicity and convenience.
The most desirable coordinate system would define thecircular arc camber line as part of a circle centered at theorigin, such that the camber line represents the portion of thecircle in the ran,-e
_r ¢ + ¢(51)2 2 2 2
when p is measured counterclockwise from the positive x axisand, from geometry,
= a1 -a2 (52)
This combined translation-rotation is accomplished by thecoordinate change
x= (x-x1 ) cos p + (y-y,) sin p (53)
y= - (x-x 1 ) sin p + (y-yl) cos p
where
p = (54)2
18
From the geometry of the camber line,
o = sin-' [(X 0 ) (55)
In the new coordinate system, then,
x = R2 (56)
In this system, either surface of the airfoil may be expressedas a portion of a circle centered on the y' axis. That is,
x + (y' _ y )2 = R 2 (57)
or, expanding,
x2 + y,2 - 2y' ys + YS R 2 (58)
In cylindrical polar coordinates, Equations (56) and (58)
may be expressed as follows
r i = R (59)
r 2 2+ y - 2 r 2 Ys sin R 2 (60)
where the relevant range of p is as given in Equation (51).
The desired thickness is the difference (Ar) between theexpressions for the two arcs. First however, Equation (60)must be solved for r2 rather than r2 . Using the binomialtheorem,
r. = Y sin -+ R 2 - 2 cos 2 (61)
If it is stipulated that Equation (61) represents the surfaceof the blade which lies further from the origin than does thecamber line, then the thickness t in a positive sense is givenby the expression r2 - r,.
19
t = ys sin R + R s - s 2 R (62)
The proper sign of the second term (+) is chosen byrequiring that at = 2,
t = Y +R - R (63)
Therefore, Equation (62) becomes
t = YS sin + R - Ys cos2 - R (64)
A further rotation of the coordinate system such that
171 (65)2 2p
yields
(66)t = y cos - 2 - ys2 sin2[ -]- R
in the range 0- ji - 4.
Expressing Equation (66) in terms of the normalized arclength, s,
(67)
t = Y cos - s ] + Rs2 - si - ) S - R
in the range 0: s< 1.
The expressions for ys and Rs in Equation (67) can bedetermined geometrically in terms of T/2 and r0, the edge radius.
R + R sin 4 (68)
S 0 sin i Arccos (-A 2
20
T (69)
where
(T!2 - to) + R (1 - cos 4/2) (70)A
R sin ¢/2
Equations (68), (69) and (70) yield all the unknownconstants of the thickness distribution in terms of knownquantities.
21
SECTION IV
CARTESIAN COORDINATES FOR THE BLADE
The two previous sections have described the methodsderived to specify individual blade sections. When located asdesired relative to the blade stacking axis, the sectioncoordinates are the coordinates of the streamsurface bladesection. A series of such sections on all given stream surfacesspecifies the envelope of the blade, but the surface coordinatesare not in a convenient form for manufacturing purposes. Amethod of determining the coordinates of plane sections throughthe blade at a series of locations along the stacking axis isnow described.
The method relies upon the use of the "spline-curve" forinterpolation (or extrapolation) of the coordinates of theblade surfaces and therefore the theory of the curve is given.The same interpolation scheme is also used elsewhere in thecomputer program.
Given the Cartesian coordinates x and y for a series ofpoints, the problem is to find a curve y = f(x) that passesthrough each given point. A series of algebraic cubics isused, one equation applying between each adjacent pair ofpoints. The four coefficients for each cubic permit thematching of the first and second derivatives of the two equationsapplying at each point (except the first and last), in additionto passing the curves through the specified points. Twoadditional c'iditions are required to establish the coefficients.Those chosen here are that the second derivatives of theequations at the first and last points are zero. Otherconditions are possible, but less appropriate for the currentpurpose. A curve of the form described is that theoreticallyassumed by a thin, uniformly-flexible beam pinned at eachspecified point, hence the name "spline-curve."
By applying the conditions described above, the followingsystem of equations may be produced. In the interval betweenany two adjacent points, such that xn l<x<xn, the equationof the spline-curve is
M1 (Xn-X)3 M (X-xn- 1 ) 3 [ Yn-1 M (6 (x n-x) +-X (x n_ x) Xn- Xn- 1 6 n n
(XXn-l Yn Mn )(71)
(x-x~1 ) _X_ 2 x-~)n 2n-2
22
4
,aid the nlope of the 8 p] ne-curve is(72)
dY M n 1 (Xn'X) 2 M n (x'x n 1)2 MY'Y- M n-1- n - + n- + __ nn- (x -X )dx 2 (x-X ) 2 (x-x 1 ) ( -x 6") n -]n-n-1 n- n-1 n- n-1
The coefficients M are [riven by Ml MN 0.0 where N is thetotal number of specified points, and
SDn-Bn M n+ (73)M TI n n--n A
n
A1 = 1.0
B1 = 0.0
D, = 0.0
(x n+1-x n) (X n-I Bn-in 3 6 A n -
A= n+l ni (x- )B
B - n IXn6
tYn+-Yn Yn'Yn_1 (xn' Xsn-) n-1
n X n+ 1 Xn X n _Xn_ 1 6 A n 1
ierce the coefficients M may be found by first computing theco-ff'lcients A, B and D.
In order to use the spline-curve for interpolation, it isconwornlent first to compute the M coefficients for all the.1pecified points, and then to evaluate the value of the function
:1 at the de-sired value (or values) of x, using the coefficientsappropriate to the interval in which x lies. Should extrapolationo)ut.:Ide the -;pecifled range of values of x be required, this is
lone a2-um~nv, a otraight line at th slope of the spline-curveat the appropriate end-point.
23
The determination of the Cartesian coordinates of themanufacturing sections involves two steps. First, Cartesiancoordinates for the points defining the streamsurfaces areobtained. Second, coordinates for the manufacturing sectionsare interpolated (or extrapolated) at the desired plane sectionsthrough the blade.
Figure 15 illustrates the geometry involved in the deter-mination of the Cartesian coordinates of the streamsurfacesections. The radius of the streamsurface is defined at anumber of axial locations. A spline-curve is passed throughthese points, and the slope of the streamsurface is calculated(as the slope of the spline-curve) at 100 points distributeduniformly on the x-axis between the first and last given points.Then an m-x table is constructed by numerically integrating forthe streamsurface using
m = m 1 + (x-x 1 ) 1 + dr + dx ) 2] 2 (74)
The m coordinate of any point on the streamsurface blade sectionis the "axial" coordinate of the point derived from the camberline and thickness distribution. Thus linear interpolation ofthe m-x table yields the corresponding x coordinate. Use ofa 100-point table assures that linear interpolation is adequate.The r-x table given enables the radius of the streamsurface atany value of x to be found. Spline-curve interpolation is usedfor this purpose.
The "y" coordinate of any point derived from the camberline and thickness distribution must now be related to thecircumferential direction on a streamsurface. This can beaccomplished by determining the angle F (,defined in Figure 15)in the following manner. First, the origin of the m-system isshifted so that the origin (m = 0) corresponds to the locationof the centroid of the section in the cascade plane. Thecamber line slopes in the cascade plane must then be reproducedon a streamsurface. This requires that at each point
dm dx , (75)
It is possible to determine the variation of e with m byintegrating Equation (75) with respect to m,
om Z (76)e(m) - co = r dm
24
II
where both the streamsurface radius and the camber line slopeare known functions of x, hence m.
Atm= 0,
Yd (77)
0 rd
where Yd is the distance between the section centroid and thecamber line measured in the y direction in the cascade plane,and rd is the streamsurface radius.
Therefore, from Equations (76) and (77),
f m ' dm 4 Yd (78)r rd
For each point on the blade surface in the cascade plane,
the appropriate 6 is determined through a six step procedure:
(1) m is determined from the m-x table
(2) ecamber line is determined from Equation (78)
(3) Ay is determined from y = ysurface - Ycamber linein the cascade plane
(4) r is determined from the value of m in conjunctionwith the m-x and r-x tables
(5) Ac is determined from Ae = Ay/r
(6) e surface camber line + As
The two remaining Cartesian coordinates of the point thenfollow from
z = r cos e (79)and
y = r sin c
The interpolation of x and y coordinates for the manufactur-ing sections (on constant-z planes) is achieved by fittingthe spline-curve through corresponding points on each stream-surface section. The same number of points is used to defineeach blade section. Hence, the spline-curve is used firstto fit x as a function of z for the first point on the suction
25
surface of each section. The value of x is then interpolated ateach desired value of z. Then this is repeated for thecoordinate y, and then for all other corresponding pointsdescribing the streamsurface sections. Thus an equal number ofsimilarly distributed points describing the manufacturingsections is obtained.
26
SECTION V
SECTION PROPERTIES
The stacking axis of the blade is passed through eachstreamsurface section either at its leading or trailing edge,or at a point specified relative to the centroid of the section.Because the streamsurface sections are in general not planar,the centroids of the manufacturing sections will not generallylie precisely on the stacking axis when the streamsurfaccsections are stacked about their centroids. By determining thelocations of the centroids of the manufacturing sections soobtained, it is possible to estimate the offsets that mustbe applied when restacking the streamsurface sections in orderthat the centroids of the manufacturing sections will be locatedas desired relative to the stacking axis. Thus there is arequirement to determine the locations of the centroids ofboth the streamsurface and manufacturing sections if the bladeis to be stacked on the centroids.
To further assist in any mechanical analysis of the blade,the areas, second moments of area, principal axes, andprincipal second moments of area for both the streamsurfaceand manufacturing sections are also determined. The methodsused to determine these quantities are given below.
1. STREAMSURFACE SECTIONS
The location of the centroid of an area relative to anarbitrary origin is given by
~fxdAfydA(80)fy dA fdA
x a n d Y' -
In the computer program these integrals are performednumerically by considering the section to be composed of asemicircle formed by the leading edge and a series ofquadrilaterals. The quadrilaterals are defined by the seriesof points defining the camber line and the values of thesection thickness at these points, as shown in Figure 16. Theorigin for the calculation is the leading edge of the blade andeach quadrilateral is assumed to be a rectangle. The lengthof one side is the camber line length enclosed by thequadrilateral, and the length of the other is the mean sectionthickness at the ends of the quadrilateral. Hence the aboverelationships are evaluated as
27
(81)
_ -__2 _ 4 ___l _,N-_ _(t--n lnt )
fi Yox1m (Sn+I-sn) 2 )(82)
IAr.o 4yo ( ) Ymn+Ymn+l1 s (tn+d A sinal + (Ynn l ) ,~
I2N-+ "2 ( n"S 2 )
7Y°2 , (t )+t 1 (83)dA= -o + (Sn+ 1-Sn) 2
1 ,N-1
For the case of the double-circular-arc blade, a termrepresenting the contribution of the trailing edge semicircle,similar to the term for the leading edge contribution, is addedto each of Equaticns (81), (82) and (83).
The second moments of area I and I, and products ofinertia, I , are found by againX considgring the section to becomposed of x. series of rectangles, as shown in Figure 16.(The contribution of the edge(s) is neglected.) The principalsecond moments of one rectange about its centroid are
3(84)tn+l +t\ /n
x (Sn+ 1 -S) 2- ) /12
I (tn+l+tn ) 3 /12 (85)ly, 2 (Sn+l-n11
The mean camber line angle, C, at the two ends of there.ctangle is used to determine the second moments of area ofthe rectangle about its centroid from
I x,+Iy I i-I , (86)
x 2 + 2 cos(2c)
I + y T I-I ,. cos(2C) (87)
2 2
28
The product of inertia of the rectangle is also determined frow
(I-l Iy) cos(2C) - Ix + 1 (88)
xy 2 sin(2C)
The contributions of the rectangle to the second moments ofarea of the section about its centroid are given by
2 (89)Alx (section centroid) Ix + A(y-)
2 (90)
A y (section centroid) I y + A(x-)
where A is the area of the rectangle, and x and y are thecoordinates of the center (that is, centroid) of the rectangle.Also, the contribution of the rectangle to the product ofinertia of the section about its centroid is given by
Al xy(section centroid) = Ixy + A(x-T) (y-Y) (91)
The summation of all such contributions yields the secondmoments of area and product of inertia of the section about itscentroid.
The inclination of the principal axes of the section, B,is fiven by
B = arctan (2
The principal second moments of area of the section about itscentroid are given by
I +1 I -i(93)= + 2 cos(2B) - I sin(2B)
1 +1 I-I (94)yp = 2 a- 2 cos(2B) + Ixy sin(2B)
29
K. . ,- : .
The various calculations described above are made for a
"normalized" streamsurface blade section (that is one having an
axial chord of unity), and are subsequently scaled as appropriate,according to the specified chord length.
2. MANUFACTURING SECTIONS
The methods used to determine the properties of themanufacturing sections are essentially the same as those describedabove for the streamsurface sections. However, in this casethe section geometry is described by the coordinates of a seriesof points on the two blade surfaces as shown in Figure 16.
The section is again considered to be composed of asemicircle at the leading edge and a number of rectangles.The radius of the leading edge, yO, is calculated to be a halfof the distance between the first points on the blade pressureand suction surfaces. The angle of the blade section at theleading edge is obtained from the first two points on eachblade surface, thus
/ YP2 +YS2 -ypl-ySl (95)=arctan K +xs2-xp1 -xs-
(xP2 +X 2-xj-i
The lengths of the sides of each assumed rectange are determinedfrom the coordinates of the points at the four corners, thus
ln T 41xn / ~l) + (YSn+l-ypn+l)2] 1 (96)= 1 1 2 ( 6V[(Xsn-XPn) + (YS n-yp+)
n n+ 1 n n+l-ySn ]+
f P[(XPnXPn)2 + (YPn+-YPn) 2
Hence, the section area and the location of the centroid aredetermined from
dA : 2 + ('nmn()
1 1
71Y 2Y 4 'yo (99)dA - [(xp 1+xsl)/2.0- 4 cosa] +
[eim 1n ( +X n+XPn+l+XSn+l)/4.0]
2 (1oo)dA = 2 [(ypl+ysl)/2. 0 - 3- Sinc] +
11n n + y n+l+YPn+y p n + l )/.0
As for the streamsurface sections, an al'litional term isinserted in each of Equations (98), (99) and (100) for thedouble-circular-arc blade, to account for the contribution ofthe trailing edge semi-circle.
The second moments of area and product of inertia of thesection are again found by first determining the principalsecond moments of each rectangle about its centroid, and thendetermining the contribution of each rectangle to the secondmoments and product of inertia of the section. Referring againto Figure 16, we have for one rectanrle
3 (101)
I = mt /12
3 (102)ly= em /12
In order to determine the second moments of area of therectangle about is centroid in directions parallel to the axisof the section coordinates, the angle of inclination of therectangle is found from
A = arctan YPn+ 1+YSri+lYnYS) (103)
XPn+l+XSn+1 x P n - xs n
Then the second moments of area and product of inertia forthte rectanile aro found using Equations (86), (87) and (88)
-alain, and the contributions of the rectang-le to the properties
31
of the sections are given by Equations (89), (90) and (91).Equations (92), (93) and (911) are applied to give the inclinationof the princioal axes, and the principal second moments of thesection about its centroid.
An additional section property, the torsional constant, iscomputed for the manufacturing sections. Following Reference 2,the constant K is determined from
K (104)
(i+4 F "
where
F = fo t 3 du (105)
and du, t, and S are the elemental length along the sectionmedian line, the thickness normal to the median line, and thesection area, respectively.
32
SECTION VI
BLADE CHARACTERISTICS
1. BLADE VOLUME CALCULATION
Except when IPRINT = 1, a calculation of the volumeenclosed by the blade between the innermost and outermoststreamsurfaces is made. This is done from the Cartesiancoordinates of the points describing the streamsurface bladesections. The total volume is obtained by summing the volumesenclosed between each adjacent pair of streamsurfaces. Thesevolumes in turn are obtained by summing two types of sub-volume. First, there is the volume enclosed by the leadingedge radius. This is determined by finding the mean area ofthe two leading edge semicircles on the two streamsurfaces,and multiplying by the mean height at the two ends of the semi-circle diameters. The diameter of the leading edge is calculatedto be the distance between the first points on the sectionpressure and suction surfaces using the streamsurface Cartesiancoordinates. Thus the areas computed are those projected ontoa constant-z plane, and the volume is given directly bymultiplying by the mean z dimension. Sccond, there is aseries of volumes enclosed by corresponding quadrilaterals onthe two streamsurfaces. Again, the sub-volume is determinedfrom the mean of the areas of the two quadrilaterals projectedonto a constant-z plane, and the mean distance between them.The quadrilaterals are each defined by two adjacent pointson the section suction surface and the two correspondingpoints on the pressure surface. The area of the quadrilateralsis obtained by assuming them to be rectangles of side t and m,as defined in Equations (96) and (97), except that in thiscase Cartesian coordinates of the streamsurface sections areused. The distance between the two rectangles is taken to bethe mean of the distance between the four corresponding cornersof the rectangle. Thus the total volume, V, is given by
(106)
-Zr Ar + nlmnl ,j+l J )(Z -Z n
l [J-l '+ i 1,N-1 2
where J is the total number of streamsurfaces, z is the meancoordinate at each end of the leading edge semicircle diameter,A is the mean of the two leading edge semicircle areas, N istfe number of points defining each blade surface, and z is themean z-coordinate at the four corners of the rectangle.n Equation(106) is modified to contain a term similar to the first torepresent the trailing edge contribution for the double-circular-arc blade.
33
2. CALCULATIONS FOR AERODYNAMIC ANALYSIS
Certain aerodynamic analyses of the blade require adescriDtion of the blade on cylindrical surfaces. As an option,the computer program will produce quantities necessary for thisdescription at a specified streamsurface-computing stationintersection.
First, the angular position of the camber line with respectto the stack axis at a streamsurface-computing station inter-section is specified in terms of D, which is the value ofScamber line on the streamsurface at that particular intersection.
The physical passage blockage (b) due to the presence ofthe blades is determined as a percentage of the passagecircumference in terms of the number of blades in the bladerow, N, and a, the angle subtended on the cylindrical surfaceby each blade.
No (107)b b = 2--
b - 27r
The blade lean angle, C, with respect to the radialdirection at a given point is obtained from the slope of aspline-iurve fit through the y-z Cartesian coordinates of thestreamsurface section camber lines at the particular axiallocation. Thus
o -Arctan(AL) (108)
Two other quantities are needed to produce the propermean-camber line angle on the cylindrical surface: the localcomputing station inclinations p, obtained from the specifiedstation description; and the local streamsurface inclination, y,obtained from the specified r-x streamsurface description. 0Together with the camber line angle on the streamsurface, a*:
tan a 0 tan y tan C + tan a*/cos y (109)
l-tan p tan y
Equation (109) is taken from Reference 3.
34
SECTION VII
USE OF THE COMPUTER PROGRAM
Basic information required by the user to run the computerprogram is given in this section. The various input data itemsare defined first, and the input data format is then specified.A description of the output data that may be expected is given.(implementation of the program on a computing system is notdiscussed here, but in the section entitled "Computer ProgramDetails".)
1. DEFINITION OF INPUT DATA ITEMS
TITLE An alphanumeric title of 72 characters that maybe used to identify a run.
NLINES The number of streamsurfaces which are definedon and on which blade sections will be designed.Must satisfy 2 < NLINES<15.
NSTNS The number of computing stations at which thestreamsurface radii are specified. Must satisfy3 < NSTNS < 10.
NZ The number of constant-z planes on whichmanufacturing (Cartesian) coordinates for theblade are required. Must satisfy 3 < NZ <15.
NSPEC The number of radially-disposed points at whichthe parameters of the blade sections arespecified. Must satisfy 1<_NSPEC<_15.
NPOINT 'De number of points that will be generated tospecify the pressure and suction surfaces ofeach blade section. Must satisfy 2<_NPOINT< 80.Generally, no less than 30 should be used. Itwill be advantageous to specify 80 points whenprecision plots of the sections are to beproduced directly by the program.
NBLADE The number of blades in the blade row.
ISTAK If ISTAK = 0, the blade will be stacked at theleading edge.
If ISTAK = 1, the blade will be stacked at thetrailing edge.
If ISTAK = 2, the blade will be stacked at, oroffset from, the section centroid.
35
IPUNCH An integer which indicates 'whether the ,]uartitIt-.necessary for aerodynamic analysiL of the bladeare to be produced on punched cards.
ISECN If ISECN = 0, the blade will be constructedusing the polynomial camber line and thestandard thickness distribution.
If ISECN = 1, the exponential camber line andthe standard thickness distribution will beused.
If ISEC11 = 2, a blade using the circular arccamber line and the double-circular-arcthickness distribution will be used.
If ISECN = 3, the otandard thicknessdistribution and any of the camber lines.described in the sub-section, "MultipleCircular-Arc Camber Lines," may be used.
IFCORD If IFCORD = 0, the meridional projections ofthe streamsurface blade section chords arespecified.
If IFCORD = 1, the streamsurface blade sectionchords are specified.
IFPLOT Where CALCOMP software is incorporated intothe computing system, IFPLOT specifies thecreation of precision plots. (Further informationregarding the requirements for this are givenin the section entitled "Computer ProgramDetails".)
If IFPLOT = 0, no plots will be producea.
If IFPLOT = 1, a plot of the streamsurfacesections will be produced. All NLINES sectionsare shown superimposed. The origin for eachsection plot is offset from the centroJd ofthe section by distances specified by DELXand DELY.
If IFPLOT = 2, a plot of the manufacturingsections will be produced. The origin is theblade stacking axis, and all NZ sections areshown superimposed.
If IFPLOT = 3, both of the plots described forIFPLOT = 1 and 2 will be produced.
36
'f" , 11'1,O)'I , irldlvlual plots of each of themariu factur in), sect Ior:; w I1 be produced. Thea;(e_ are rotated clockwise by the section.'Aa)-f,er anfrle for each plot.
'7l H ipnpt data I:; alway.s listed by the program.DetalIl; of the stream:;urface and, manufacturing'."ctions are, printed a:7 prescribpd by IPRINT.
rf Ipp i T = 0, details of streamsurface and
,-anifacturnr sections are printed.
If I!RINT = 1, detalls of streamsurface;.ect1ons are printed.
if IPRINT = 2, details of the manufacturing_ectIonns are printed.
?'.Z:INU , Te NZ manufacturing sections are equispaced.OUTI'PR between z equals ZINNER and ZOUTER.
'CALE When precision plots are produced, SCALE isthe scale factor employed.
.2TTACKX This is the axial coordinate of the stackingaxis for the blade, relative to the sameorigin as used for the station locations, XSTA.
PLTSZE qe size (inches) of the plotter to be usedin the creation of precision plots.
KPTS The number of points provided to specify theshape of a computing station.
If KPTS = 1, the computing station is uprightand linear.
If KPTS = 2, the computing station is linearand either upright or inclined.
If KPTS > 2, a spline curve is fit through thepoints provided to specify the shape of thestation.
IFANGS If IFANGS = 0, the calculations of thequantities required for aerodynamic analysiswill be omitted at a particular computingstation.
If IFANGS = 1, these calculations will beperformed at that station.
37
XSTA An array of KPTS axial coordinates (relative toan arbitrary origin) which, together with RSTA,specify the shape of a particular computingstation.
RSTA An array of KPTS radii which, together withXSTA, specify the shape of a parcicularcomputing station.
R The streamsurface radii at NLINES locationsat each of the NSTNS stations.
ZR The variation of properties of the streamsurfaceblade sections is specified as a function ofstreamsurface number. The various quantitiesare then interpolated (or extrapolated) ateach streamsurface. The streamsurfaces arenumbered consecutively from the innermostoutward, starting with 1.0. ZR must increasemonotonically, there being 1SPEC values in all.
Bl The blade inlet angle. (Figure 17 illustratesthe geometry of a typical blade section.)
B2 The blade outlet angle.
PP If ISECN = 0, PP is the ratio of the secondderivative of the camber line at the leadingedge to its maximum value. Must satisfy-2.0 < PP < 1.0.
If ISECN = 1, PP is the ratio of the secondderivative of the camber line at the leadingedge to its maximum value forward of theinflection point. Must satisfy 0.0 < PP < 1.0.
If ISECN = 2 or 3, PP is superfluous.
QQ If ISECN = 0, QQ is the ratio of the secondderivative of the camber line at the trailingedge to its maximum value. Must satisfy0 .0 :5 QQ :5 1. 0.
If ISECN = 1, QO is the ratio of the secondderivative of the camber line at tie trailingedge to its maximum value rearward of theinflection point. Must satisfy 0.0 < QQ S 1.0.
NOTE: Some comments regarding useful values forPP, QQ, S and BS are made in the sectionentitled "Examples of Use of Computer Program."
38
If ISECN = 2 or 3, QQ is superfluous.
RLE The ratio of blade leading edge radius tochord.
TC The ratio of blade maximum thickness to chord.
TE The ratio of blade trailing edge half-thicknessto chord.
If ISECN = 2, TE is superfluous.
Z The location of the blade maximum thickness,as a fraction of camber line length.
If ISECN = 2, Z is superfluous.
CORD If IFCORD = 0, CORD is the meridional projectionof the blade chord.
If IFCORD = 1, CORD is the blade chord.
DELX, The stacking axis passes through the stream-DELY surface blade sections, offset from the
centroids, leading, or trailing edge by DELXand DELY in the x and y directions respectively.
S, BS If ISECN = 1 or 3, S and BS are used to specifythe location of the inflection point (as afraction of the meridionally-projected chordlength) and the change in camber angle fromthe leading edge to the inflection point.If the absolute value of the angle at theinflection point is larger than the absolutevalue of Bl, BS should have the same sign asBl; otherwise, Bl and BS should be of oppositesign.
2. INPUT DATA FORMAT
Data input is by punched card, and three formats are used.The first card only is alphanumeric, using the first 72 columnson the card. This is followed by one card of integers. Integersare each placed in a field of three locations, starting inColumn 1 of the card. No decimal point may be used, and thenumber must occupy the right-most locations of the allocatedfield. Real numbers are each placed in fields of 12 locations,starting in Column 1 of the card. Decimal. points should beincluded to ensure correct interpretation of the data, and thenumbers may be placed anywhere in the allocated fields.
39
In the following chart, one line corresponds to one card.
TITLE
NLINES NSTNS NZ NSPEC NPOINT NBLADE ISTAK IPUNCH ISECN IFCORD IFPLOT IPRINT
ZINNER ZOUTER SCALE STACKX PLTSZE
KPTS IFANGS
XSTA RSTA f repeated KPTS times rrepeated NSTNS timesR repeated NLINES times
ZR Bl B2 PP QQ RLE
TC TE Z CORD DELX DELY repeated NSPEC times
S BS - if ISECN = I or 3 only
Listings of sample input data decks are included under"Examples of Use of Computer Program."
3. OUTPUT DATA
Printed output from the program may be considered toconsist of four sections: a printout of the input data, detailsof the blade sections on each streamsurface, a listing ofquantities required for aerodynamic analysis, and details of themanufacturing sections determined on the constant-z planes. Theseare briefly described below. In the explanation which follows,parenthetical statements are understood to refer to the particularcase of the double-circular-arc blade (ISECN = 2).
The input data printout includes all quantities read in,and is self-explanatory.
Details of the streamsurface blade sections are printed ifIPRINT = 0 or 1. Listed first are the parameters defining theblade section. These are interpolated at the streamsurface fromthe tables read in. Then follow details of the blade section in"normalized" form. The blade section geometry is given for thesection specified, except that the meridional project of thechord is unity. For this section of the output, the coordinateorigin is the blade leading edge. The following quantities aregiven: blade chord; stagger angle; camber angle; section area;location of centroid of the section; second moments of area ofthe section about the centroid; orientation of the principalaxes; and the principal second moments of area of the sectionabout the centroid. Then are listed the coordinates of thecamber line, the camber line angle, the section thickness, andthe coordinates of the blade surfaces. NPOINT values are given.
40
_ . . . _ ., _. , -, m -r n - r I i n -i|I I I ! i.
A lineprinter plot of the normalized section follows. The scalesfor the plot are arranged so that the section just fills the page,so that the scales will generally differ from one plot to another."Dimensional" details of the blade section are given next. Thenormalized data given previously is scaled to give a bladesection as defined by IFCORD and CORD. For this section of theoutput, the coordinates are with respect to the blade stackingaxis. The following quantities are given: blade chord; radiusand location of center of leading (and trailing) edge(s);section area; the second moments of area of the section about thecentroid; and the principal second moments of area of thesection about the centroid. The coordinates of NPOINT pointson the blade surfaces are then listed, followed by thecoordinates of 31 points distributed at (roughly) six degreeintervals around the leading (and trailing) edges. Finally,the coordinates of the blade surfaces and points around theleading (and trailing) edge(s) is (are) shown in Cartesianform.
The quantities required for aerodynamic analysis areprinted at all computing stations specified by the IFANGSparameter. The radius, section angle, blade lean angle, bladeblockage, and relative angular location of the camber line areprinted at each streamsurface intersection with the particularcomputing station.
Details of the manufacturing sections are printed ifIPRINT = 0 or 2. At each value of z specified by ZINNER,ZOUTER and NZ, section properties and coordinates are given.The origin for the coordinates is the blade stacking axis.The following quantities are given: section area; the locationof the centroid of the section; the second moments of area ofthe section about the centroid; the principal second moments ofarea of the section about the centroid; the orientation of theprincipal axes; and the section torsional constant. Thenthe coordinates of NPOINT points on the blade section surfacesare listed, followed by 31 points around the leading (andtrailing) edge(s).
Precision plots are produced if IFPLOT = 1, 2, 3 or 4 asdescribed under the definition of IFPLOT given previously.
If IPUNCH = 1, the program punches the quantities requiredfor aerodynamic analysis, together with identifying indicesdenoting station number and streamsurface number, on cards inthe following format: 5 fields each of 12 locations for thequantities themselves, followed by 2 fields each of 3 locationsfor the indices.
41
SECTION VIII
EXAMPLES OF USE OF THE PROGRAM
I. IPTRODUCTION
This section shows the use of the program to generate fourblades: one based upon the polynomial camber line; anotherupon the exponential camber line; a double-circular-arc blade;and one based upon the multiple-circular-arc camber line.The same basic aerodynamic design is assumed for the first twoand last cases, although all blades could not satisfy the sameaerodynamic requirements. Figure 18 is a meridional sectionthrough the compressor annulus, showing eight streamsurfaces(including the hub and casing) defining the flow, and the locationsof six planes on which manufacturing sections are to be determined.Usually, more streamlines would be used to define the flow andthus the blade, by means of the streamsurface blade sections.Also, it would normally be advantageous to determine more thansix manufacturing sections. However, these numbers were selectedfor the sample runs of the program as they exemplify the programadequately and lend clarity to the resulting figures. Includedon Figure 18 are the approximate locations of the blade leadingand trailing edges for the polynomial camber line case. Theexponential and multiple-circular-arc camber line cases wouldappear slightly different for reasons disciosed below.
2. POLYNOMIAL CAMBER LINE
a. Input Data
The input data deck used for this example is listedbelow. Some points of interest are noted in the order in whichthey occur in the input.
As mentioned previously, eight streamsurface bladesections are used to define the blade. The streamsurface radiiare specified at eight locations, the first and last of whichare outside, and well clear of, the rotor. This ensures thatthe boundary conditions imposed on the spline-curve (zerocurvature at the end points) have little influence on the shapeof the curve representing the streamsurface in the region ofinterest, that is, within the blade. The parameters definingthe streamsurface blade sections are given at eight locations;that is, directly at each streamsurface. Thus the interpolationof the parameters at each streamsurface is reduced in thiscase to merely reading the appropriate row of data from thetable. Fifty points are specified to define each blade surface.This number serves the purposes of this example well, but itwould be advantageous to use more points where the precision-plot output is to be directly incorporated in the manufacturing
42
procedure. Specification of the meridionally-projected chord(rather than actual chord) is oelected. All optional section'sof the output are to be printed. Superimposed plot.; of the!streamurface and manufacturing sections are to be produc,(i ata ,scale of five times full size.
Blade inlet and outlet angles would no~mally bedetermined from an aerodynamic analysis of the flow throuvh treblade row. Desirable incidence angles for this profile typeare probably near to one half the leading edge wedge angle.Considerations of static pressure (or velocity) dlstrlbutiors.within the blade channel, and choking flow limits, will influencethe selection of incidence angles, depending upon the Zlectedaerodynamic analysis method. Deviation angles for the )rofilehave not been empirically determined as yet, and therefore thebest means of estimating deviation angles is probably to useCarter's Rule, or a similar formulation. Some consideration ofthe camber line shape (point of maximum camber, for examile)should be made. The parameters PP and QQ are set to 0.0 and 0.5on all streamsurfaces, respectively. It is anticipated thit thezero curvature leading edge configuration (PP = 0.0) will btfrequently specified for this profile type. The specificationof a second derivative at the trailing edge that is less thanits maximum value (QQ = 0.5) is made to prevent the curvatureof the camber line from rising to a maximum at the trailingedge.. It is believed that this could cause excessive deviationangles. The blade thickness distxPibution of a rotor blade isdetermined by aerodynamic and mechanical factors. The leadingedge radius and trailing edge half-thickness are set toapproximately .005 inch, which is probably a practical minimum.Maximum thickness/chord ratios specified vary from 11.8% at thehub to 3.1% at the casing. These figures are typical of thoseaerodynamically acceptable for a high Mach number rotor blade,and yield a blade with a reasonable distribution of cross-sectional area. The maximum thickness is placed at 0.6 of thecamber line from the leading edge, which helps to maintain asmall leading edge wedge angle.
43
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b. Output Data
The input data specified all optional output data,and a selection for each section of the output is shown below.Some points of interest are as follows.
Shown first is the printout of the input data; allinput items are listed. This is followed by details of thefirst streamsurface blade section. The parameters defining theblade section are followed by details of, and a line-printerplot of, the normalized blade section. Then appears aspecification of the section scaled to the desired dimensions.First there are the fifty points specified for each bladesurface. These are printed two per line, and if an odd numberof points is specified, coordinates of the last point are notprinted. Next follow the coordinates of the 31 points describingthe leading edge radius. The final data for the streamsurfacesection are the equivalent Cartesian coordinates for the pointsjust mentioned. It will be seen that the first and last pointson the leading edge radius coincide with the first points onthe pressure and suction surfaces.
The same format is repeated for each of the remainingstreamsurface blade sections, and a printout of the volumeenclosed by the blade, but these results have not beenreproduced here. The calculations for aerodynamic analysis areprinted next for those computng stations for which they wererequested by the IFANGS parameter, presented in order ofincreasing radial streamsurface location.
Details of the six manufacturing sections definingthe blade follow. Reproduced below is the output relating tothe first (innermost) section. Properties of the sectionare followed by the coordinates of 50 points on each bladesurface, and 31 points around the leading edge. It will benoted that the section centroid is not calculated to beexactly on the stacking axis (the origin for the coordinates)although the offsets DELX and DELY were all zero. This is areflection of the fact that the streamsurface sections isnot planar. If it were desired to have the centroids of themanufacturing sections lie precisely on the stacking axis,the program would be rerun with DELX and DELY offsetsspecified so that they counteract the mislocation of thecentroids previously determined. This iterative proceduremight require more than one loop, depending upon the accuracydesired. Use of the centroid offsets alo permits the finalblade to be leaned for stress redistribution or other purposes.
The input data specified superimposed precision plotsof both the streamsurface sections and manufacturing sections.These plots are reproduced (at reduced size) as Figures 19 and
416
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20, respectively. It is of interest to refer also to Figure 18.The innermost manufacturing plane is well below the lowest pointon the hub streamsurface section in the plane of the stackingaxis, that is, the plane of Figure 18. The Cartesian coordinatesof the first streamsurface blade section show that the lowestpoint on the section (the 20th point on the leading edge) is atz = 6.66478. The radius of the streamsurface at this point is6.75 (approximately), and the innermost manufacturing plane isat z = 6.5. Thus, at the leading edge the extrapolationrequired to define the manufacturing section is somewhat smallerthan might at first appear. At the trailing edge, extrapolationis required to define the first two manufacturing sections.The streamsurface radius at the casing and z-coordinate forthe outermost manufacturing plane both equal 9.0 at the bladeinlet. However, the z-coordinate of the leading edge of theblade is 8.8357, so that the outermost manufacturing section toois defined completely by extrapolation. Of course, portionsof the blade that are defined by extrapolation would notappear on a final blade, but would facilitate manufacture.
47
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3. EXPONENTIAL CAMBER LINE
a. Input Data
As mentioned previously, the overall design incorporatedin the polynomial camber line example was also assumed for theexponential camber line example. The discussion of the inputdata therefore highlights the changes made to the polynomialcamber line example data in order to create the exponential camberline example.
The exponential camber line is specified by settingISECN equal to 1. For this example, chords rather thanmeridionally-projected chords are specified. The chord lengthsused are taken from the results of the polynomial camber lineexample, and because of the different camber line configurations,the resulting meridionally-projected chords are somewhatsmaller for the exponential camber line case. The parameters PPand QQ are set equal to 0.5 for all sections. These valueswill probably be satisfactory for most applications of theblade profile. The only remaining change to the polynomialcamber line input data is to specify the distributions of theinflection point and change in camber angle from blade inletto the inflection point. The distributions used for theexample are quite arbitrary, but representative of what mightbe indicated by a typical aerodynamic analysis. The inflectionpoint is located at the leading edge on the innermost stream-surface section, and moved rearward .05 per streamsurface, to•35 (of the meridionally-projected chord) at the casing.(As a value of s of zero is not acceptable, the value used forthe innermost section is .01.) For the three innermostsections, the inflection angle is set equal to the inlet angle.Thus, these sections have straight leading edge portions (oflength from zero to .1 of the meridionally-projected chord),followed by portions of continuously-positive camber. Theremaining sections have inflection angles that are greater thanthe section inlet angles, the difference in angles rising fromzero at the third section (as described above) to seven degreesat the casing. Thus, these sections have reverse and positivecamber, the reverse camber increasing with radius, as theinflection point also moves rearward.
The input data deck is listed below.
62
FIGURE EXAMPLE 8LADE DESIGN - EXPONENTIAL CAMBER LINE8 8 6 8 51 3, 2 L 1 1 3 0
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65
b. Output Data
The input calls for all optional printed output. Asthis is identical in format to that shown for the polynomialcamber line example, none has been reproduced here.
Figures 21 and 22 show (at reduced size) the super-imposed streamsurface section and manufacturing section plots.The comments regarding extrapolation made in connection withthe polynomial camber line example again apply. A comparisonof some intermediate sections from both plots shows well thevalue of an accurate method of determining manufacturingcoordinate data from streamsurface design data. For bladeswhere the streamsurface section form is critical throughoutthe section, any approximate method based upon blade edgegeometry only is unlikely to be satisfactory.
11. DOUBLE-CIRCULAR-ARC BLADE
a. Input Data
This example illustrates several program features notillustrated by the previous examples. First, this exampleillustrates curvilinear computing stations. Secondly, theblade is described in terms of negative inlet angle, illustratingthat the program accommodates blades specified in either sense.This allows stators to be specified in the opposite sense torotors, if desired. Finally, the blade is stacked at itstrailing edge, rather than at, or offset slightly from, thecentroid.
The double-circular-arc blade is specified by settingISECN equal to 2. The particular example illustrates a noveldesign concept in that the leading edge is swept from hub andtip toward mid-passage, a concept which is described in furtherdetail in Reference 4.
The input data is listed below.
66
FIGURE EXAMPLE CASE - DOUBLE CIRCULAR ARC BLADE8 5 5 8 5U 49 I L 2 1 3 2
7.5 8o625 5,0 5. 129,'8 i2,78796 7*6272012,9748u 7,74720u3,1152Jj 7.8728uo3.2U12hu 8.00450u3.2241UL 8.142LO03e1746o, 8,2862Gu3.04u66j 8.4392662.79 4uu 8,61146u7,6272Ju7.7472uu7.87286-8.145.ju
8.2862iu80 439201,8.6114JL8 13,34u925 7,7326dt3,48116u 7.83348U3.58b64J 7.94u9813.b5vgj9 8,u5 49483.668u75 8.1744383.63695. 8.2989953.3u,2u 8.4281763.3428J. 8.5638j67.7326ou7@83348u7994L9818* 5431. 88.1744388.2989958.4281768,5638us8 13.89395- 7.7467iU3.974j, 7.8473294. 576ju 7.9553324.1ou6ju d.J699744.11265L 8,1895284.u873dL 8,3i3284,G2U3jL 8.4394583.89523u 8.5692007.7467J67.8473297.9553328.069974
67
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8,1942768. 325 9998,457 4338* 5891'j..
0tu -45.895. 8#4951 .3021a Go.,21 s 5u t 2.0349-1
2ttJ 45.7b5L 90~UU50 * 0023.t;43%; 6323 o5u,.t. 2.143J
3 -i -4i o 25U 9 .5L. 50 sjG25*u4'uj qu.J25 05ULO 1*9903
4 ! 4*5a 9. 895U *j0260 .au u26 .5G0) 1.8953
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6 0.*, i -46. 775t, iu*215L, *0026*u0 UG26 a~uGO 1*9230
7 * &u uu -48.135L 10.2150 9J024.0uo024 *5%ojj 2.0720
.0J uu2i *5uUu 2o3650
68
b. Output Data
The printed output for this example is similar in mostrespects to that shown for the polynomial camber line. Thesection entitled "Dimensional Results" is shown for stream-surface section 1 to illustrate that the section coordinatesare presented relative to the stack axis which, in this case,has been specified at the blade trailing edge. Two other minordifferences in comparison to the polynomial camber line outputare related to the trailing edge: The printout contains thecoordinate location of the center of the trailing edge as wellas the 31 points distributed around the trailing edge. Thelatter feature also appears with the printout of the manu-facturing sections (not reproduced here).
Figures 23 and 24 show (at reduced size) streamsurfacesection and manufacturing section plots, respectively. Thecomments regarding extrapolation made in connection with thepolynomial camber line example again apply.
69
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5. MULTIPLE-CIRCULAR-ARC CAMBER LINE
a. Input Data
The same overall design incorporated in the firsttwo examples was also assumed for the multiple-circular-arccamber line example. The discussion of the input data will belimited to changes made to the exponential camber line exampleto create the multiple-circular-arc camber line example.
The multiple-circular-arc camber line is specified bysetting ISECN equal to 3. The only other changes made to theexponential camber line example are the distributions ofinflection point and the change in camber angle from bladeinlet to the inflection point. The inflection point has beenarbitrarily set at 0.3 (of the meridionally-projected chord)on all streamsurfaces. The change in camber angle has beenspecified in such a way that the blade has positive camber inthe leading segment at the hub, gradually decreasing to zerocamber (straight line leading segment) on streamsurface 4, andproducing sections with reverse camber on the four outermoststreamsurfaces.
The input data deck is listed below.
74
FIGUFE PULTIPLE CIRCULAR ARC CAMBER LINE8 8 6 8 5=C 30 2 0 3 1 3 C
6.5 9.0 5.0 0.0 t2.01 c;
-1.4357 0.C696236.974E7.31977,66!8.01268.36818*735C9.12
2 C-1°6357 0,0-1,02E4 6.756.757.068 c
7.38447.69948.015 c
89336589662.9.0
1 1-. 5767 0.060886147916347o442(7.7259
8.016S8.316C8.62338°9382
1 1
-,2087 0.0E.98727.233E7.484 .
7,7421
75
8.0061
8.2777
1 1o20003 0.070,78E?.295C7.51647.743E7.977c8.22018.47328.7345
1 1*6093 0.07oi'4967933877.532E7,73257.94038,15838.38as8o631,
2 C2o0620 .C1026S 7.20767.2O7E7.375c7.54SE7.73017,918S8911978.337789584E
i c1.427" 0.07.27iE7*.42147.576e
76
7.738(7. 90 548.0805=
8.2651
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0.3 1.80
4.0 -62.686 -i.965 0.0 0.5 .00163.03970 oCI163 0.6 1993560.3 0.05.s0 -403.48 -23.758 0s 0 0.5 .00159903731 900159 0.6 1. 87710.3 -2*1316.0 -E6.623 -28.327 0.0 0.5 *00155o 03503 .0015r, 0.6 1. 81870.3 -5,4257.0 -E•.471 -34.37 0.0 0.5 .00153.03317 OC015=3 0.E 1.76030.3 -10.628.0 -67.1 -42.52 0.0 0.5 a00150.03129 CCI15c 0.6 1970180.3 -18*141
77
kLi
b. Output Data
The input data calls for all optional printed output.As this is identical in format to that shown for the polynomialcmboer line, none has been reproduced here.
Figures 25 and 26 show (at reduced size) the super-imposed streamsurface section and manufacturing section plots,respectively. The comments regarding extrapolation made inconnection with the polynomial camber line example are againapplicable.
78
SECTION IX
COMPUTER PROGRAM DETAILS
1. IMPLEMENTATION OF THE COMPUTER PROGRAM
The program is written in FORTRAN IV and was developed onan IBM 7094/7044 Direct Couple System (incorporating theversion 13 IBSYS Operating System) and a CDC 6000 SeriesSystem (incorporating the version 3.3 SCOPE Operating System).When loaded into core, the program (and resident system)occupies about 32K of storage, so that the program will probablynot be usable without modification on a relatively smallcomputer. Apart from this limitation, the program should becompatible with the majority of modern computing systems. Theprogram consists of a main program, and Subroutines BQ, CQ, Dl,EQ, FQ and GQ. The seven decks have been given the identifiersA, B, C, D, E, F and G, respectively. Listings of the decksare shown below, and the deck set-up required for the CDC 6000Series System is also presented.
The program uses three numerical system units for itsinput and output routines. Input is drawn from the card unitvia READ statements referring to Unit LOG 1; output is sent tothe line printer by WRITE statements referring to Unit LOG 2;and punched output is produced via WRITE statements referencedto Unit LOG 3. Units LOG 1, LOG 2, and LOG 3 are set equal to5, 6, and 7, respectively on cards A1140-60 in the FORTRANprogramming. On the CDC 6000 Series System, the "PROGRAM" cardmust also establish tL.he input and output linkages. On othercomputing systems, the "PROGRAM" card may not be required, andthe input-output files may be established via control cards.
The progran as presented herein utilizes an on-lineprecision plotting capability available at the programdevelopment site. Calls to four subroutines not included inthe deck are included in the program. These calls are executedonly if precision plots are specified in the input data, and theentry points expressed are part of standard CALCOMP softwarenormally supplied to users of CALCOMP precision plottingequipment. The various call statements used in the programare explained below, to facilitate modification should the needarise.
CALL PLOT (XPLOT, YPLOT, N)
The majority of plotting is done using this form of call. Theparameters XPLOT and YPLOT are the "x" and "y" coordinates (ininches) on the paper to which the pen is being directed. Theparameter N indicates pen up or down, INI = 3 or INI = 2respectively, and will cause XPLOT or YPLOT to be assigned
79
as the origin for further coordinates if N is negative.
CALL SYMBOL (X, Y, H, TEXT, THETA, N)
This call is used to title the plots. The parameters X and Yare the coordinates (in inches) of the lower left hand cornerof the first character, H is the character height (in inches),TEXT is the character to be printed, THETA is the angle of thelettering with respect to the "x" axis and N is the totalnumber of characters to be printed.
CALL NUMBER (X, Y, H, F, THETA, N)
This call causes the printing of the number F. The parametersX, Y, H, and THETA are used as for CALL SYMBOL. The parameterN indicates the number of digits following the decimal pointif positive, or truncation to an integer if equal to -1.
CALL PLOTE
This call terminates the tape.
In the event the orogram is used on a computing systemwhich does not include CALCOMP software, and the operatingsystem will not execute a program with unsatisfied externalreferences, dummy entry points may be supplied by adding tothe deck a subroutine such as the following:
SUBROUTINE PLOT
A= A
ENTRY SYMBOL
ENTRY NUMBER
ENTRY PLOTE
RE TURN
END
2. DECK SETUP FOR CDC 6000 SERIES SYSTEM
The deck setup required to run the program on a CDC 6000Series System incorporating the SCOPE 3.3 Operating System issho'.!n below. Production runs of the program would usuallyemploy relocatable binary forms of the routines producedfrom the source decks to avoid having to compile the FORTRANfor each run and the waste of associated computer resources.
80
JOB Identification, etc.
FTN.
LGO.
7/8/9 End of Record
SOURCE DECK A
SOURCE DECK B
SOURCE DECK C
SOURCE DECK D
SOURCE DECK E
SOURCE DECK F
SOURCE DECK G
7/8/9 End of Record
DATA DECK
6/7/8/9 End of Job
3. FORTRAN PROGRAM LISTING
A listing of the FORTRAN program appears on the followingpages with each subroutine started on a new page.
81
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123
4. PROGRAM LOGIC
The analysis which has been described in this report isperformed primarily in the main program and Subroutine BQ,while the other subroutine performs specific tasks supportiveof the desired objectives. Subroutine BQ concerns the detailsof defining and describing the blade on the streamsurface; themain program utilizes this information to stack the blade andobtain the manufacturing section description. Subroutine CQis used to find slopes of various "spline-curves" at particularpoints. Dl is the curve-fitting routine, EQ produces a plotin the printed portion of the output, and FQ is used to drawstreamsurface and/or manufacturing section plot axes if IFPLOT =1, 2 or 3. Subroutine GQ is used to compute the center coordinate,the radius of curvature, and the arc length for the circular-arc and multiple-circul.ar-arc camber lines.
A dscription of the calculation procedure employed inthe btain program and in Subroutine BQ is described below.The steps of this procedure are keyed to their location in theprogram through the associated deck serialization, which appearsparenthetically.
1. The input data is read and printed. (A1190-A2160)
2. If precision plotting is specified, the plot is initializedand if IFPLOT = 1 or 3, the axes for the superimposed stream-surface plots are produced. (A2170-A2200)
3. A loop performed for each streamsurface section is commenced.This loop creates the blade on the streamsurface. (A221.0)
11. The axial locations of the intersections of a particularstreamsurface with all computing stations are determined, theinterval between the first and last points is subdivided into99 intervals, and the axial coordinates of the 100 resultinglocations obtained. (A2220-A2290)
5. The slope of the streamsurface at each of the 100 locationsobtained in 4. above is derived. (A2300)
6. The slope of the computing stations at each of the stream-surface computing station intersection points is derived.(A2310)
7. Using Equation (74), the streamsurface length (in themeridional projection) is obtained over the 99 intervals of4., providing an x-m table for the streamsurface. (A2320-A2340)
8. The parameters defining the streamsurface blade section areinterpolated (or extrapolated) from the input tables. IfNSPEC = 1, they are taken to be radially uniform. If NSPEC 2,
124
linear interpolation is employed. If NSPEC >2, spline-curveinterpolation is used. (A2350-A2500)
9. The design of the streamsurface section is initiated.(A2510-A254o)
10. If IPRINT = 0 or 1, the parameters defining the stream-surface blade section are printed. (B1230-BI410)
11. If ISECN = 0, the coefficients of the quartic camber lineare computed using Equations (6), (7), (8), (9) and (10).The chord length corresponding to a meridional camber linechord of unity is determined. (B1450-B1510)
12. If ISECN = 1, the coefficients for the two segments of theexponential camber line are computed using Equations (17), (18),(19), (20), (21), (22), (23) and (24). Equations (18) and (24)require iterative solution, and up to 50 attempts to satisfythe equations are permitted. The equations are taken to besatisfied when the parameter P or 9, as appropriate, is givento within 0.0001 of its specified value. (Should the iterativeprocedure fail, a diagnostic is printed and the job is terminated.This will not normally occur.) The chord length correspondingto a meridional camber line chord of unity is determined.(B1530-B2050)
13. If ISECN = 3, the constants of the equations for the twosegments of the multiple-circular-arc camber line are determinedfrom Equations (38), (36), (35), and (34) respectively, inSubroutine GQ. These constants are scaled and the arc centeroffset, if required, and the arc length of each segment determinedcumulatively. The chord length corresponding to a meridionalcamber line chord of unity is determined. (B2070-B2300)
NOTE: Each segment is described by a proportion of the NPOINTcamber line points based on its proportion of the meridionallyprojected chord length. If this number of points for theleading segment is 0 or 1, this segment is abandoned, andone circular arc is produced for the entire camber line.
11. If ISECN = 2, the constants of the equation for thecircular-arc camber line and the camber line arc length arecomputed as for ISECN = 3. The chord length corresponding to ameridional chord of unity, that corresponding to an overallblade chord of unity, and an associated scale factor(described in 15.) are determined for the double-circular-arcblade. (B2320-B2350)
15. For ISECN = 0, 1 or 3, the chord length corresponding toan overall blade chord of unity as well as a scaling factorwhich relates the ratio of dimensions for an overall meridionalchord of unity to dimensions for a camber line meridional
125
chord of unity are determined. (B2370-B2380)
16. If IFCORD = 1, the specified chord is divided by the chordcorresponding to a unity meridional chord, giving the desiredmeridional chord. The three blade thickness descriptors arescaled so that they apply to a blade having a camber linemeridional chord of unity. (B2390-B2420)
17. The coefficients of the two thickness equations of thestandard thickness distribution are computed using Equations(42), (43), (114), (47), (48) and (49). (B2470-B2520)
18. The length of the section camber line for ISECN = 0 or 1is determined by numerical integration. The meridional chord(of unity) is divided into (NPOINT-I) uniform intervals, whereNPOINT is the number of points specified in the input to defineeach blade surface. The integration for the curve lengthbetween each adjacent pair of points is obtained using Simpson'sRule, the interval being subdivided into 10 equal intervals.If the polynomial camber line is specified, the gradient ofthe camber line is given by Equation (4). If the exponentialcamber line is specified, the gradient of the camber line isgiven by Equation (15), different coefficients applying forwardand rearward of the inflection point. These data enable alocation on the camber line to be expressed as a fraction ofcamber line length. (B2540-B2730)
19. At each of the NPOINT points, equispaced on the meridionalchord, the section half-thickness is computed from Equations(41), (46) or (67), and the coordinate and slope of the camberline are computed, from either Equations (5) and (4) (polynomial),(16) and (15) (exponential), or (26) and the derivative of (26)solved for (x-x ), choosing the appropriate branch of the0square root involved, for the circular-arc and multiple-circular-arc camber lines. (B2970-B31170)
20. The coordinates of NPOINT points on each blade surface aredetermined by combining the data derived in 19. The scalingfactor derived in 14. or 15. is applied so that the bladecoordinates correspond to an overall blade meridional chord ofunity. (B3180-B3520)
21. The slope of the camber line is expressed in degrees. Thecamber line coordinates, lengths, and the section half-thicknessare scaled by the factor derived in 14. or 15. (B3530-B3570)
22. The section area and the location of the centroid aredetermined using Equations (81), (82) and (83). (B3580-B3730)
23. The camber line is redefined in terms of 100 points toassure sufficient points for accurate linear interpolationsin the determination of e, needed for the eventual trans-
126
formation onto manufacturing sections. (B3780-B4240)
24. The scaled camber line is interpolated to yield itsdescription over 99 equal axial intervals. (B'1250-B4260)
25. The camber line y-coordinate and slope at the axiallocation of the centroid or centroid-offset are determined bylinear interpolation. (B4270-B4280)
26. The product of inertia and second moments of area of thesection about the centroid are determined using Equations (84),(85), (86), (87), (88), (89), (90) and (91). (B4310-B4470)
27. The orientation of the principal axes and the principalsecond moments of area of the section about the centroid aredetermined using Equations (92), (93) and (94). (B4480-B4520)
28. If IPRINT = 0 or 1, details of the normalized blade section(meridional chord unity) are printed. (B4570-B4850)
29. If IPRINT = 0 or 1, a line-printer plot of the section ismade. (B4870-B4940)
30. Section properties determined in 22., 26., and 27. arescaled according to the specified section chord to produce"dimensional" results. (B4950-B5120)
31. If IPRINT = 0 or 1, rection properties are printed.(B5150-B5420)
32. The coordinates of the NPOINT points on the blade surfacesare scaled according to the specified chord, and the origin isshifted to the stacking axis. The stacking axis is offsetfrom the centroid of the section by the specified distances.If IPRINT = 0 or 1, the coordinates are printed. (B5510-B5670)
33. The coordinates of 31 points describing the blade edge(s)are determined. For the standard thickness distribution, it isassumed that the wedge angle is zero, so that the radiusextends over a complete semicircle, hence the points aredistributed at 60 intervals. For the double-circular-arcblade, the 2nd and 30th points of the edges are determined atthe exact location of the tangency between the particular bladesurface and the edge radius, resulting in an angular intervaldistribution uniform between points 2 and 30, with irregularfirst and last intervals. (B5740-B6220)
34. The origin of the m-coordinate system is shifted to thesection centroid or centroid-offset as provided by the inputdata. (B6250-B6290)
127
K:-I, i I Iii -
35. The m-length corresponding to each of the 100 pointsdescribing the camber line is determined. (B6360-B6410)
36. The reference e0 of Equation (77) is determined. (B6420)
37. The integration of Equation (78) is performed toward theends of the blade section from the centroid or centroid-offset,giving the c coordinate for the camber line. (B6430-B6840)
38. The m-coordinate of each point of the blade surface isdetermined. (B6870-B6980)
39. The c coordinate system is shifted to the leading ortrailing edge if the blade is stacked there. (B7000-B7030)
40. The e corresponding to each point on the section surfaceis determined from the e of the camber line at that axiallocation and Ae calculated from the displacement from thecamber line and the streamsurface radius. (B7070-B7260)
411. The location of the stacking axis is given (in the inputdata) by specifying its axial coordinate. Linear interpolationyields the corresponding streamsurface coordinate from the x-mtable created in 7. (A2550)
42. The origin of the m-coordinate shifted in 34. is againshifted, to the stacking axis (determined in 41.). (A2560-A2580)
43. The origin of the table of axial coordinates of thestreamsurface-computing station intersection points (obtainedin 4.) is shifted to the stacking axis. (A2590-A2600)
411. If IFPLOT = 1 or 3, the streamsurface blade section plotis produced. (A2620-A2850)
45. A trigger is set if the calculation of quantities for aero-dynamic analysis of the blade is specified at any computingstation. (A2860-A2890)
46. If the calculations for aerodynamic analysis are specified,the angular location of the camber line with respect to thestack axis, the slope of the camber line on the streamsurface,and the Cartesian coordinates of the camber line at the stream-surface-computing station intersections are determined andstored. (A2910-A2960)
47. The axial coordinate of each of the NPOINT points specifyingthe streamsurface blade section suction surface is obtainedby linear, interpolation from the x-m table produced in 42.(A2970-A2990)
128
48. The streamsurface radius at each axial location derivedin 47. is obtained by spline-curve interpolation from the tablecreated in 43. (A3000)
49. Cartesian coordinates for each point on the streamsurfacesection suction surface are obtained using the 's's computedin 40. and Equation (79). (A3010-A3060)
50. Steps 47., 48., and 49. are repeated for the pressuresurface. (A3070-A3160)
51. Steps 47., 48., and 49. are repeated for the pointsdescribing the leading edge. (A3170-A3260)
52. Steps 47., 48., and 119. are repeated for the pointsdescribing the trailing edge if ISECN = 2. (A3280-A3370)
53. If IPRINT = 0 or I, the Cartesian coordinates of eachpoint describing the streamsurface blade section are printed.(A3390-A3910)
54. The loop initiated in 3. for each streamsurface isterminated. Upon completion of this loop, the Cartesiancoordinates of all streamsurface sections have now been storedfor interpolation on the specified manufacturing planes.(A3920)
55. Unless IPRINT = 1, the volume of the blade is calculated,using Equation (106), and printed. (A3950-AI4130)
56. If the calculations for aerodynamic analysis are specified,the program prints the appropriate heading. If not, theprogram moves to step 62. (A4160-A4180)
57. In the original x-r coordinate system, the axial coordinateof the streamsurfaces-computing station intersections and thecomputing station slope, are determined for each station atwhich the aerodynamic analysis quantities were specified.(A4260-A4270)
58. The blade blockage of Equation (107) is obtained at eachof these points. (A4280-A4400)
59. The blade lean angle of Equation (108) is obtained at eachof these points. (A4510)
60. The mean section angle of Equation (109) is obtained ateach of these points. (A4530-A4540)
61. The quantities required for aerodynamic analysis areprinted, and produced on cards if IPUNCH = 1. (A4550-A4570)
129
K
62. If IFPLOT = 2 or 3, the axes are drawn and titled for thesuperimposed plot of all the manufacturing sections. (A4620-A4630)
63. If no output relating to manufacturing sections isspecified by either IFPLOT or IPRINT, the remainder of theprogram is bypassed. Alternatively, if printed details of themanufacturing sections are specified, a heading is printed.(A4640-A4700)
64. The location of each of the manufacturing planes isdetermined. (A4710-A4760)
65. The (Cartesian) coordinates of each of NPOINT points onthe blade surface are obtained by spline-curve interpolationat each of the manufacturing sections. (A4770-A4860)
66. The (Cartesian) coordinates of each of 31 points describingthe blade edge(s) are obtained by spline-curve interpolationat each of the manufacturing sections. (A4870-A4990)
67. A loop that is performed for each manufacturing sectionis initiated. This loop contains the determination of sectionproperties and the output of results for the section. (A5000)
68. The section area and location of the centroid aredetermined using Equations (98), (99) and (100). (A5010-A5250)
69. The torsional constant is determined using Equations (104)and (105). (A5260-A5420)
70. The second moments of area and product of inertia for themanufacturing section about its centroid are determined usingEquations (101), (102), (103), (86), (87), (88), (89), (90)and (91). (A5430-A5650)
71. From the second moments of area and product of inertia,the orientation of the principal axes and hence principalmoments of area are determined, using Equations (92), (93)and (94). (A5660-A5690)
72. If IPRINT = 0 or 2, section properties and coordinatesare printed. (A5720-A6260)
73. If IFPLOT = 2 or 3, a plot of the manufacturing section isproduced. (A6310-A6530)
74. If IFPLOT = 4, an individual plot of the manufacturingsections is made. The axes are rotated clockwise until thechord line is horizontal. The angle of rotation is indicatedas the stagger angle. (A6550-A7070)
130
75. The loop initiated in Step 67. for each manufacturingsection is terminated. (A7080)
76. If precision plots have been made, the plotting isterminated. (A7090)
131
V I Iia
0.8-
0.7-
CCRPH = 0.000.- 0.500
0.3-
0.2-
0.1-
0.0 -T I5 -0.0 0.1 0.2 0.3 0.4~ 0.5 0.6 0.7 0.8 0.9 1.0x
Figure 1 Variation of Normalized Second Derivativeof' Polynomial Camber Line with Parameter tP?
132
1.3-
1.2-
1.1
1.0-
0.9-
0.8-RLPHR1 60.00
0.7- RLPHR2 =30.00a =:0.500
0.6- :=0.000p: 0.250
0.5- p 0.500
0. L-
0.3-
0.2-
0.1-
0.0*0.0 0.1 0.2 0.3 0.14 0.5 0.6 0.7 0.8 0.9 1.0
xt Figure 2 Variation of Polynomial Camber Line with
Parameter P
133
0.9-
0.8-
0.7-
0.6-LPR=550
0.5- .0p = 0.000
0.4- p = 0.250p= 0.500
0.3-
0.2-
0.1-
0.0 I
0.0 0.1 0.2 0.3 0.14 0.5 0.6 0.7 0.8 0.9 1.0x
Figure 3 Variation of Polynomial Camber Line withParameter 'P'
134
1.3-
1.2-
1.0-
0.9-
0.8-RLPHRI 60.00
0.7- RLPHA2 =30.00p : 0.000
0.6- Q = 0.500a = 0.750
0.5- a = 1.000
0.4-
0.3-
0.2-
0.1-
0.0 -----0.0 0.1 0.2 0.3 0.14 0.5 0.6 0.7 0:8 0.9 1.0x
Figure 4~ Variation of Polynomial camber Line withParameter IQ'
135
1.5-
1.4
1.3-
1.2-
1.1
1.0-
0.9-
0.8-RLPHIIR= 60.000.8000
0.7- .0
FiLPIIR2 =10.00
0.6- FLPHR2 =30.00RLPH-A2 =50.00
0.5-
0.4-
0.3-
0.2-
0.1-
0.0 0.1 0.2 0.3 0.4~ 0.5 0.6 0.7 0.8 0.9 1.0x
Figure 5 variation of Polynomial Camber Line with
Alpha 2
136
0.6-
ALPHAI = 60.000.6 ALPHAS = 67.00
RLPHR2 = '40.000.4- S = 0.300
P = O = 0.3000.2- P Q = 0.500
P = : = 0.700
0.o
> -0.2
-0.4
-0.6
-0.8
0.0 0.1 0.2 0.3 0.LI 0.5 0.6 0.7 0.8 0.9 1.0X
Figure 6 Variation of Normalized Second Derivativeof Exponential Camber Line with Parameters'P' and IQ'
137
1.8
1.7
1.6
1.5
1.4-
1.3
1.2
1.1
1.0
>- 0.9 RLPHA1 60.00RLPHRS - 67.00
0.8 RLPHR2 4 '0.00
0.7- S 0.300P = 0 0.300
0.6 P 0 :0.5000.5P = 0.700
0.5-
0.3
0.2
0.1
0.0-0.0 0.2 0.4 0.6 0.8 1.0
X
Figure 1 Variation of Exponential Camber Line withParameters 'P' and 'Q'
138
2.1
2.0-
1.9-
1.8-
1.7-
1.6-
1.5-
1.4-
1.3-
1.2-
1.1 FLPHA1 =60.00
1.0- ALPiAS = 67.000.9 RLPHA2 =60.00
0.-S = 0.7000. =o :=0.100
0.7- PQ = : =0.3000.6- -PQ : =0.700
0.6-0.5-
0.3-
0.2-
0.1
0.0 I I I I I
0.0 0.2 0.14 0.6 0.8 1.0x
Figure 8 Variation of Exponential camber Line with
Parameters IP' and IQ'
139
2.0-
1.9
1.8
1.7
1.6
1.3
1.2
I. 1. ALPHRI = 60.00
1.01 ALPHRS = 67.00
0.9 ALPHR2 : 60.00P = 0 = 0.3005 = 0.300
0.7 s - 0.5000.6 5 : 0.700
0.5-
0.4-
0.3
0.2
0.1
0.0 . . ..
0.0 0.2 0.4I 0.6 0.8 1.0x
, I •~ , I
1.6
1.5
1.4- ALPHRI = 60.00RLPHAS = 60.00
1.3 ALPHR2 = 40.00P - = 0.300
1.2 3 = 0.010S = 0.250
11 5 = 0.500
1.0,
0.9/
0.8
0.6
0.5
0.-
0.3
0.2
0.1
0.00.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
x
'- , i~ ,,r f j Fxr, ,*rntlal Camber Line with
RLrMI1 = 10.00ROMR~S = 10.00RLOMR2 = 10.00S = *0. 300
RLFMIIR = 30.00RLI'IRS = 30.00RLFI-R2 =10.00
RLFUR1 60.00RLI'tRS =30.00RLFI-R2 = 10.00S = 0.300
RLFMI =~ 10.00RLOMRS = 40.00RLPJIR2 = 10.00
Figure 11 Some Camber Lines Available Under theMultiple-Circular-Arc Camber Line Option
14l2
(0
cr.CM
RLPHRI = 60.00RLPHAS = 67.00RLPHR2 = 60.00S = 0.300
OS s = 0.500
0o S = 0.700
(D 0!
ct.
0
0.2 0.4 0.6 O.B 1.0x
Figure 12 Variation of Multiple-Circular-Arc CamberLine with Parameter 'S'
143
RLPHRl = 60.00A PLPHRS = 60.00RLPHR2 40.00S =0.00
S = 050
C,.00 020 0. 40 0.0 ..\10
0x
Fo/
.144
e
00.00 0.20 0.40 0.60 0.80 1.0
Figure 13 Variation of Multiple-Circular-Arc CamberLine with Parameter 'S
11411
YZERO = 0.00250T = 0.05000TONE = 0.00250Z 0.70000
YZE19O = 0.00250T =0.05000TONE =0.00250
z = 0.60000
YZERO =0.00250T =0.05000TONE =0.00250
Z = 0.50000
Figure 14I Variation of' Thickness Distribution withPoint of' Maximum Thickness
14l5
m
y z
x
Typical Point
Figure 15 Cartesian and Streamsurpace Coordinates
of a Point
14~6
STREANSURFACE BLADE SECTION
y
xtn n+1
xrn ELEMENTAL AREA, ASSUMEDn________ RECTANGULAR
x
MANUFACTURING BLADE SECTION
y
ELEMENTAL AREA, ________
ASSUMED RECTANGULAR
x
Figure 16 Calculation of' Section Properties
l17
SS
AC
SYMBOL DESCRIPTION RELATED INPUT DATA ITEM
a1 Inlet Angle Bi
a2 Outlet Angle B2
YO Leading Edge Radius RLE = Y0 /C
T Maximum Thickness TC T/C
Yi Trailing Edge Half-Thickness TE yY/C
Z Location of Maximum Thickness Z Z (Camberline Length)
C Chord CORD if IFCORD = 1
AC Axially-Projected Chord CORD if IFCORD =0
S Inflection Point S = s/AC
a Inflection Point Angle BS + BI
Figure 17 Parameters Describing Blade Sections
148
86
75
6
54
STREAMSURFACES MANUFACTURING4 3 PLANES
3
2
LEADING EDGE OF BLADE TRAILING EDGE OF BLADE
Figure 18 Locations of Streamsurfaces and ManufacturingPlanes for Example Blade Designs
149
Streamsurface Sections
Figure 19 Example Blade Design - PolynomialCamber Line
150
Cartesian Sections
Figure 20 Example Blade Design - PolynomialCamber Line
151
Streamsurf'ace Sections
Figure 21 Example Blade Design - ExponentialCamber Line
152
/q
//
Cartesian Sections
Figure 22 Example Blade Design - ExponentialCamber Line
153
St ream-- urfac-- '7,ctiono~
Figure 23 Example Blade Design - Double-Circular-Arc Blade
154
Fiprure 2hj Example Blade Design -Double-Circular-
Arc Blade
155
Streamsurface Sections
Figure 25 Example Blade Design - Multiple-Circular-Arc Camber Line
156
Cartesian Sections
Figure 26 Example Blade Design - Multiple-Circular-Arc Camber Line
157
REFERENCES
1. Hearsey, R. M., and Wennerstrom, A. J., "Axial CompressorAirfoils for Supersonic Mach Numbers", Aerospace ResearchLaboratories, Wright-Patterson AFB, Ohio, ARL 70-0046,March 1970.
2. Roark, Raymond J., Formulas for Stress and Strain, McGraw-Hill. (New York), 1954.
3. Tysl, E. R., Schwenk, F. C., and Watkins, T. T., "ExperimentalInvestigation of a Transonic Compressor Rotor with a 1.5-InchChord Length and an Aspect Ratio of 3.0 I-Design, Over-allPerformance, and Rotating-Stall Characteristics", NACA RM E54L31,National Advisory Committee for Aeronautics, 1955.
4. Wennerstrom, A. J., and Hearsey, R. M., "The Design of anAxial Compressor Stage for a Total Pressure Ratio of 3 to 1",Aerospace Research Laboratories, Wright-Patterson AFB, Ohio,ARL 71-0061, March 1971.
2.58
r . .. .