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A M E R I C A N
C a p 7 Z
S T A N D A R D
Letter Symbo ls for
Aeronaut ica l Sciences
ASA Y10.7-1954 UDC 003.62:533.6
Prepared by Sectional Committee on
Letter Symbols
The American Society of Mechanical Engineers
National Advisory Committee for Aeronautics
Institute of the Aeronautical Sciences
American Rocket Society
T H E A M E R I C A N S O C I E T Y O F M E C H A N I C A L E N G I N E E R S 29 West 39th Street, New York 18, N. Y.
T A B L E OF CONTENTS
General Pr inc ip les o f L e t t e r Symbol Standardization ....................................... L e t t e r Symbols f o r Primary C o n c e p t s ................................................................ L e t t e r Symbols for Secondary C o n c e p t s ...........................................................
Symbols for u s e a s Subscripts .......................................................................... Symbols for u s e a s Superscr ipts ......................................................................
............................................................................................................ Appendix I Recommended Des igna t ions for Sys tems of Axes, Angular Relat ion-
s h i p s , and Quanti t ies R e l a t e d There to
Appendix 11 .............................................................................................................. Alphabet ical Index b y C o n c e p t s
P a g e
5
Sectional Committee on Letter Symbols H.M. Turner, Chairm.an, A s s o c i a t e Professor , E lec t r ica l Engineering,
Y a l e Universi ty , New Yaven, Conn.
P e r s o n n e l of Subcommittee No. 7 on Le t te r Symbols for Aeronaut ical S c i e n c e s
Thomas F. Ball, Chairman, Bureau of Ordnance, Department of the Navy, Washington, D.C. Roy C. Hopgood, Secretary, 420 Lexington Ave., New York, N.Y.. J.V. Charyk, Department of Aeronautical Engineering, Princeton University, Princeton, N. J. Charles W. Chillson, Curtiss-Wright Aeronautical Corporation, Caldwell, N.J. W.M. Coates, U.S. Naval Post Graduate School, Monterey, Calif. W.S. Diehl, Captain, USN, Bureau of Aeronautics, Department of the Navy, Washington, D.C. Hugh L. Dryden, Director of Aeronautical Research, National Advisory Committee for Aeronautics,
Washington, D.C. Donne11 W. Dutton, Director, Daniel Guggenheim School of Aeronautics, Georgia Institute of Technology,
Atlanta, Ga. Maurice A. Garbell, President, Maurice A. Garbell, Inq., San Francisco, Calif. 1.E. Garrick, National Advisory Committee for Aeronautics, Langley Aeronautical Laboratory,
Langley Field, Hampton, Va. Newman A. Hall, Mechanical Engineering Department, University of Minnesota, Minneapolis, Minn. James A. Hootman, National Advisory Committee for Aeronautics, Washington,, D.C. H.C. Johnson, Glenn L. Martin Co., Baltimore, Md. C.T.G. Looney, Department of Civil Engineering, Yale University, New Haven, Conn. Clark B. Millikan, Director, Guggenheim Aeronautical Laboratory, California Institute of Technology,
Pasadena, Calif. F.T. Morse, Department of Mechanical Engineering, University of Virginia, Chartlottesville, Va. John D. Nicolaides, Bureau of Ordnance, Department of the Navy, Washington, D.C. R. Hosmer Norris, General Engineering Laboratory, General Electric Co., Schnectady, N.Y. G.L. Shue, Consolidated Vultee Aircraft Corporation, San Diego Division, San Diego, Calif. Harold K. Skramstad, Naval Ordnance Laboratory-Corona, Corona, Calif. M.J. Thompson, Defense Research Laboratory, University of Texas, Austin, Tex. R.J. Volluz, Ordnance Aerophysics Laboratory, Consolidated Vultee Aircraft Corporation,
Daingerfield, Texas Calvin N. Warfield, Applied Phys ics Laboratory, The Johns Hopk~ns University, Silver Spring, Md. John M. Wuerth, Aerophysics Laboratory, North American Aviation Corporation, Los Angeles, Calif. Charles H. Zimmerman, National Advisory Committee for Aeronautics, Langley Aeronautical Laboratory,
Langley Field, Hampton, Va.
Copyright, 1954, by THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS
Printed in U.S.A.
Foreword (This Foreword i s not a part of the American Standard Let ter Symbols for Aeronautical Sc iences , Y10.7-1954.)
This standard was prepared by Subcommittee No. 7 on Aeronautical Sciences, one of fifteen subcommittees appointed by Sectional Committee Y 10 on Letter Symbols, for the purpose of prepar- ing American Standard letter symbols. The other subcommittees cover mathematics, hydraulics, mechanics of solid bodies, structural analysis, heat and thermodynamics, illuminating engineer- ing, electrical and magnetic quantities, radio, physics, chemical engineering, acoustics, mete- orology, feedback-control systems, rocket propul- sion, and general principles.
Sectional Committee Y10 (originally desig- nated Z10) was organized under the procedure of the American Standards Association in January, 1926, with the American Society of Civil En- gineers, American Institute of Electrical En- gineers, American Society for Engineering Edu- cation, American Society of Mechanical Engineers, and the American Association for the Advance- ment of Science,, a s joint sponsors. The commit- t ee was reorganized in 1935, in 1947, and igain in April, 1951, with ASME sole sponsor. It now consis ts of representatives of 42 national socie- t ies and associations.
Subcommittee No. 7 was originally constituted in April, 1926, and submitted a proposal which was issued a s a standard in 1930. Rapid develop- ments in concepts relating to the aeronautical sciences rendered the 1930 standard obsolete, and the subcommittee was reconstituted in June, 1947, with substantially i t s present membership, for the purpose of developing a new standard. A proposed standard was published in January, 1949, for trial and criticism, and the resulting American Standard in this field was published in January, 1950.
The 1950 standard was admittedly deficient in certain respects, for compromises had been made in order to expedite approval and publication. Since the 1950publication, the subcommittee has attempted to rationalize some of the original in- consistencies, and the re quired number of changes has appeared to justify reissue of the standard in i t s present revised form. Concepts beyond those listed in the present revised standard are the subject of further subcommittee work, con- templating a possible later revision.
During the course of i ts work the subcommittee
has considered a number of special letter symbols relating particularly to meteorology, to feedback- control systems, and to jet propulsion (including rockem).* I t has been the consensus that some , - - - ~~
let ter symbols and concepts in these fields are s o specialized a s not properly to fall within the scope of Subcommittee No. 7. Special Y10 sub- committees have now been established in these fields, and therefore no attempt h a s been made to be exhaustive (in the present standard) in the matter of letter symbols for special concepts in meteorology, in feedback-control systems, and in internal aspects of propulsion.
Recognizing the practical fact that most p u b lished and duplicated papers and other works in the fields of aeronautics and aerodynamics are duplicated by some process relying upon the use of a typewriter, the subcommittee has made no u s e of type-face variations in orderto distinguish le t ter symbols. I t i s believed that this feature will simplify thepreparatibn of typewritten dupli- cations, but, of course, General Principle 4 (see page 6) will st i l l be applicable to printed publi- cations.
In the present standard there are two listings of the same symbols and quantities (in the case of letter symbols for primary concepts and for secondary concepts) in order to facilitate use of the standard. The first l ist ing i s alphabetical by symbol, and the second i s alphabetical by con- cept (see Appendix 11, page 23). The first two columns (namely, the Symbol column and the Concept column) of the first l ist ing constitute the American Standard, and all other columns and listings (including Appendix Ion axes, angu- l a r relationships, and related quantities) are merely to facilitate use of the standard. In the third column of the first l ist ing of letter symbols for primary concepts, illustrative dimensions are given, but these dimensions are not a part of the standard; in the fourth column of the first listing the extent of agreement with current American Standards is indicated and i t will be noted that this agreement i s substantial; in the fifth column of the first listing are certain remarks including definitions, but these form no part of the standard.
'Pending recommendations by other ASA-Y10 subcommit- teen on tbeae topica, aome aymbola and concepts in theae fields a m recommended herein.
In general, any of the recommended let ter s y m bols for primary concepts may b e employed a s subscripts i n application to another symbol for a primary concept. This recognizes the fac t that a given primary concept may in certain circum- s tances be employed a s a modifying or secondary concept. F o r this reason, there h a s been no deliberate attempt made to include in the sub- scr ip t l i s t a l l the symbols which may be employed a s subscripts, particularly where such'symbols maybe found in the published l is t ing o r principal le t te r symbols. In certain c a s e s , recommended subscripts may be used a s superscripts.
Multiple subscripts and superscripts a re fre- quently necessary in the more complex phases of t he aeronautical sc iences and are condoned provided the following system i s observed:
(a) Two adjacent subscripts (superscripts) impiy that the second subscript (super- script) modifies the first subscript (super- script) or that both subscripts together represent an abbreviation (e.g. z, , - position of aerodynamic center).
(b) Two adjacent subscripts (superscripts) separated by a comma imply that t he second modifies the quantity represented by the principal symbol.
(c) A subscript to a subscript denotes t h e derivative of the quantity, represented by the principal symbol, with respect to the quantity related to the second subscript.
Thus, for example:
~ C L '~ ,denoteh 7 , and
ck!
a s subscripts whenever the terms occur in mathe- matical expressions; e.g., be-elevator span; S,- wing area. Capital let ters should be used to des- ignate component parts of a configuration, a s used in models for wind-tunnel testing, and various com- ponents of the model under t e s t should be distin- guished by means of numer~ca l sub scripts; thus, F,B,N,would represent a model including the No. 1 fuselage, the No. 3 wing, and the No. 2 nacelle. Since such configuration symbols are not usually employed a s subscr ip ts , they are not included in the present l i s t of subscripts; thus, in the present l is t of subscripts , symbols for aircraft components generally appear a s lower-case letters.
In the field of turbulent-flow analysis , practice i s not well enough established to permit the recommendation of symbols pertaining to turbu- lence intensity, to turbulence energy distribution (i. e., the so-called spectrum of turbulence), and to turbulent-fluctuation frequency. Meanwhillt, the limited needs for such symbols should be met by use 'of symbols for analogous concepts in other fields. I t will be noted, however, that recom- mendations are made herein for symbols to ex- press mean-velocity component (E) and turbulent- velocity' component (u ' ) , the recommended sym- bols being the bar and the prime, respectively, to be applied to a suitable velocity symbol.
Since publication of ASA Standard 210.7-1950, the subcommittee has given careful consideration to the subjects of axis conventions (with attend- an t problems of moment designations, coeffi- cients, etc.), number designations, and flutter. Much i s now standardized in these fields, and Appendix I contains further recommendations on the first of these t o ~ i c s .
dCl denotes - A draft, dated April, 1954, was presented to the members of Sectional Committee Y 1 0 for letter ($1 ballot vote. Following its approval and that of the sponsor, i t was submitted to the American
In designating aircraft components, i t i s recom- Standards Association for approval and designa- mended that lower-case let ters be general1 y used tion. T h i s mas granted on 0 ctober 7, 1954.
Any part of this standard may be quoted. Credit l ines shoutd read: "Extracted from American Standant Let ter Symbols for Aeronautical Sciences (ASA Y10.7-1954) wi th the pennisswn of the publisher, The American Society of Mechanical Engineers, 29 U'est 39th Street, New York 18, N.Y."
AMERICAN STANDARD
LETTER SYMBOLS FOR AERONAUTICAL SCIENCES
General Principles of
Letter Symbol Standardization
1 Letter Symbols. A letter symbol for a phy si- cal quantity 1s a single letter, specified a s to general form of type. It i s available for use within 3 mathematical expression. This primary symbol may be modified by subscript or superscript. In a published work, the same primary letter symbol should appear throughout for the same generic physical quantity, regardless of the units em- ployed, and of special values assigned.
Usually, one may readily distinguish between letter symbols for physical quantities, and other published symbols and signs. The latter, while more or less analogous, belong largely to topics not treated here, such as , (i) mathematical signs, (ii) chemical symbols, (iii) punctuation, (iv) proofreading signs, and (v) abbreviations (widely used in tabulations).
2 Four General Requirements for Published Symbols and Signs. A s i s generally recognized, each published symbol or sigfl, of whatever kind, should be a t least:
(a) Standard, where ~ o s s i b l e . In the use of published symbols, authors of technical works (including textbooks) are urged to adopt the symbols in this and other current standard l is ts , and to conform to the principles stated here. An author should give a table of the symbols he uses and their respective interpretations, or e l se refer to a standard l ist a s source for symbols which he i s using but does pot explain. For work in a specialized or developing field, the author may need symbols in addition to those already contained in standard lists. In such a case he should be careful to select simple suggestive symbols which yet avoid conflict in the given field and in other closely related special fields. Except in this situation, he should not introduce new symbols or depart from currently accepted notation.
(b) Clear in reference. One should not assign to a given symbol different meanings in such a manner as to make i t s interpretation in a given context ambiguous. Conflicts must be avoided. Often a listed alternative symbol or a modi+ing subscript i s available and should be adopted. Except in brief reports, any symbol not
familiar to the reading public should have i t s meaning defined in the text. The units should be indicated whenever necessary.
( c ) Easily 'identified. Because of the many numerals, letters and signs which are sirni- lar in appearance, a writer should be careful in calling for separate symbols which in published form might be confused by the reader. For ex- ample, many letters in the Greek alphabets (lower case and capital) are practically indis- tinguishable from English letters; the zero i s easily mistaken for a capital (A
(d) Economical in ~ubl icat ion. One should try to keep a t a minimum the cost of publishing symbols. In particular: ( i) Notations which call for handsetting of movable type should be rejected in favor of forms adapted to modem mechanical methods of composition. (ii) No one work should use a great variety of types and special characters. (iii) Handwriting of inserted symbols, in copy largely typewritten and to be reproduced in facsimile, should not be excessive. (iv) Often a complicated expression appears as component part of a complex mathematical f o r mula, for example, a s an exponent of a given base. Instead, one may introduce, locally, a single letter to stand for such a complicated component. An explanatory definition should then appear in the immediate context.
3 Secondary Symbols. Subscripts and super- scripts are widely used and for a variety of con- ventional purposes. For example, a subscript may indicate: (i) the place of a term in a se- quence or matrix, (ii) a designated state, point, part, or time, or system of units, (iii) the con- stancy of one independent physical quantity among others on which a given quantity depends for i ts value, (iv) a variable with respect to which the given quantity, is a derivative. Like- wise, for example, a superscript may indicate: (i) the exponent for a power,(ii) a distinguishing label, ( i i i) a unit or (iv) a tensor index. The intended sense must be clear in each case. Several subscripts or superscripts sometimes separated by commas may be attached to a single letter. A symbol with a superscript such a s a prime (') or second ("), or a tensor index,
October, 1954 5
AMERICAN STANDARD
should be enclosed in parentheses, braces or brackets before an exponent i s attached. So far as logical clarity permits, one should avoid attaching subscripts and superscripts to sub- scripts and superscripts. Abbreviations, them- selves standardized, may appear among sub- scripts. A conventional sign, or abbreviation, indicating the adopted unit may be attached to a letter symbol, or corresponding numeral. Refer- ence marks, such a s numbers in distinctive type, may be attached to words and abbreviations, but not to letter svmbols.
4 Typography. Letter symbols for physical quantities, and their subscripts and superscripts, whether upper case, lower case, or in small capitals, when appearing a s light-face letters of
the English alphabet, are printed in italic (sloping) type. Arabic numerals, and letters of other alphabets used in mathematical expressions are normally printed in vertical type. When a special alphabet is required, boldface type i s to be preferred to German, Gothic or Script tvpe. In material t6 be reproduced in facsimile, tram copy largely typewritten, letters which would be bold- face in print, may be indicated to be such by special underscoring, while the few distinct let- ters used from other alphabets, if carefully made, should be self-explanatory. It i s important to select a type face which has italic forms, and with clearly distinguished upper case, lower case, and small capitals. Only type faces with serifs are recommended.
Letter Symbols for Aeronautical Sciences A symbol designated as an alternate isnot preferred. Symbolshaving equal rank are not designated a s alternates. The dimensions given in the third column and the definitions given in the fifth column are merely illustrative and
form no part of the present standard. With regard to the indicated dimensions, and in a particular system of units, M repre- sents Mass, L represents Length,. T represents Time, 'and 8 represents Temperature.
Symbol
Ai (written together)
b b
Letter
Concept
Slope of lift curve
Velocity of sound Area, cross-sectional
Aspect ratio
Aspect ratio, alternate for
Blade width (propellers) Semichord, alternate for
c/2 (flutter)
Chord, of an airfoil
Section coefficient Section coefficient
(flutter)
Symbols for Primary Concepts
Dimensionless
Dimensions
Dimensionless
Use c for alternate Use S for surface or
projected area See (written together)
for alternate
Also Appears in ASA Publication
Dimensionless Dimens'ionless
Remarks
L Used in Z10.3,Z10.6, 1 210.8 for breadth
L
In flutter, section coefficients reduce to steady-flow coef- ficients under steady-flow conditions. Use subscripts or superscripts to indicate any or all of the following items:
Use s for alternate for semispan (b/2)
Use b as alternate for c/2, semichord (flutter)
(1) whether for lift or moment,
(2) acting on what component or about what axis,
(3) due to what kind of mo- tion (or degree of free- dom),
(4) of what component. These items may be indi- cated in the form c ::
h w 34 (e.g. c l o ); since c ,, is , in general, a complex quantity, i t can be divided into its real and imaginary Darts. as:-
AMERICAN STANDARD
L E T T E R SYMBOLS FOR PRIMARY CONCEPTS (Cont.)
Symbol
Velocity of sound, alter- nate for; g a s velocity (rockets)
Specific heat, a t constant
Concept
pressure Specific heat, at constant
volume Circulation function, for
harmonic oscillations ( flutter)
Also Appears Dimensions in ASA Publication
Coefficient
Concentration I Diameter
D i arne ter
Drag
Span effectiveness Internal energy, per unit
mass or per unit weight, alternate for
Energy
Modulus of elasticity; Young's modulus
2T-28-1
L2T-20-1
Dimensionless
Dimensionless
Dimensionless
Remarks
Used in expressing a function, as C(k), meaning circula- tion function of reduced frequency k (see NACA Report 496 by Theodorsen). C(k) = F(k) + iG(k), or, in terms of Hankel functions,
See Appendix I and Figure 2; also, see K for factor which may have dimensions
Interchangeable, but con- flicts with differential operators shall be avoided
Dimensionless L ~ T - ~ (per unit
mass); L (per unit weight)
M L ~ T - ~
ML' IT-2
Cyclic frequency
Force; resultant force
Thrust; stream thrust (combus tion)
Acceleration due to gravity
S tructur a1 damping coefficient (flutter)
LT-2
Dimensionles s
Distinguish from u for circular frequency
See Appendix I
Use T for propeller thrust
Use with appropriate subscript to indicate type of vibra- tion. Section damping force or moment may be introduced proportional to amplitude and in phase with velocity, as , for example, by applying the factor(1tig) to the stiffness term
L E T T E R SYMBOLS FOR AERONAUTICAL SCIENCES
Mass velocity; mass- flow, per unit cross- sectional area, per unit time; weight-flow, per unit cross-sectional area, per unit time
Shear modulus
Symbol
ML-'T- (mass flow)
ML-'T'~ (weight flow)
M L - ~ T - ~
A 1 ti tude Blade thickness
(propellers) Deflection, from a speci-
fied plane, due to bending (flutter)
Concept
Enthalpy, per unit mass or per unit weight: specific enthalpy
Heat-flow rate, per unit of area, per degree, across a boundary surf ace
Angul ar mom en tum
Dimensions
Enthalpy; total heat con tent
Angle of incidence
Also Appears in ASA Publication
Impulse
Remarks
Mass moment of inertia Mass moment of inertia,
per unit length Area moment of inertia
L ~ T - ~ (per unit mass)
er unit weight)
M T - ~ O - '
I
Dimensionless I
Function of space coordinates and time; use z (lc) for alternate, when with refer- ence to z-y plane
Usually characterized by a sub script
Subscripts should be used to connote type of impulse
Advance ratio of propeller
Mechanical equivalent of heat; Joule's constant
Torsional st iffness con stan t
Dimensionless
Dimm sionless Numerical value is a function of mechanical and thermal units used
Used in torsional stiffness GI; same a s area polar moment of inertia only for a circular cross-section
Radius of gyration k I Ratio of specific heats,
alternate for Reduced frequency
(flutter) Thermal conductivity Factor, for comparison
purposes
Length; distance
Li ft Lift, per unit length
(flutter)
L 1 Z10.3,Z10.6,Z10.8 Dimensionless 210.4
Dimensionless ob "% 'P
M L T ' ~ ~ -' May have
dimensions
L
MLT-2 MT-2
210.4,Z10.6,ZlO. 12
ZIO.3,Z10.4,21O.6 Recommend useof italicized symbol, and script ell (lower case) if italicized symbol i s not available.
AMERICAN STANDARD
LETTER SYMBOLS FOR.PRIMARY CONCEPTS (Cod.)
Remarks
Use NMo for alternate
See Appendix I; see also Q
See Appendix I Usually characterized by
subscript; see Appendix I1
See Appendix I; use ox
for alternate
See Appendix I; use o for alternate
v2 PT
See also
See Appendix I; use oZ for alternate
Always used with appropriate subscript
Identification i s to be made by suitable subscripts, superscripts, or indication of functional dependence
A s occuringin pv =RT
Also Appears in ASA Publication
Z10.2,Z10.3,Zl0.4, Z10.6,ZlO. 12
Z10.3,Z10.8
~10.2,Z10.3,Zl0.6,Z10.12
Z10.2,Z10.3,Z10.4,Z10.12
Z10.2,Z10.3,Z10.4,Zl0.6, Z10.8,Z~10.12 Z10.2,Z10.3,Z10.6,Z10.12
210.4
Z10.4,ZlO. 12
'210.4,Z10.6,ZlO. 12
Z10.2,Z10.3,Z10.4,210.6, Z10.8,Z10.12
210.4,Z10.6,ZlO. 12
Dimensions
L
M
ML" Dimensionless
M L ~ T - ~
MLT'~
Dimensionless
T"
M L T ' ~ Dimensionless
T'l
ML"T'~
M L ~ T ' ~
T"
ML-'T '~ ML2T'3
L'T-~ (per unit mass) (. (per unit wei9 t )
M L ~ T ' M L ~ T - ~
T ' ~
L
Dimensionless
Dimensionless
L 2 ~ - 2 0 - 1
L
Symbol
L
rn
rn M M M
n
n
N N
P
P
P
9
4 9
9
Q Q
r
r
r
R
R R
Concept
Turbulence scale; correlation length (turbulence)
Mass
Mass,per unitlength(f1utter) Mach number
Moment, torcpe
Moment, per unit length (flutter)
Load factor
Revolutions per unit time
Normal force, alternate for Number, in general
Angular velocity of body- axis system about X-axis
Pressure; static pressure
Power
Angular velocity of body- axis system about Y-axis
Dynamic pressure Quantity of heat, per unit
time Quantity of heat,.per unit
mass or per unlt weight; specific quantity of heat
Quantity of heat Torque (propellers and
rotary-wing aircraft)
Angular velocity of body- axis system about Z-axis
Radius
Reduced radius of gyration (flutter)
Correlation coefficient (turbulence)
Gas constant Range
LETTER SYMBOLS FOR AERONAUTICAL SCIENCES
LETTER SYMBOLS FOR PRIMARY CONCEPTS (Cont.)
Symbol I Concept
Entropy, per unit mass or per unit weight; specif ic entropy
R R
/ Semispan, alternate for S
Entropy s Area, surface or projected
Reynolds number Richness; equivalence
ratio (combustion)
t Thickness t Time
t
T I Temperature, absolute
Temperature, general
Thrust (propellers and rotary-wing aircraft)
Internal energy, per unit mass or per unlt weight specif ic internal energy
Velocity, component of, along X-axis
Internal energy Heat transfer coefficient,
over-all Velocity, alternate for;
gust velocity
Velocity, component of, along Y-axis
Volume, per unit mass or per unit weight; specif ic volume
1 Velocity, resultant; speed
Dimensions
V
Dimensionless ~ i m e n s ionless
Volume
L ~ T - ~ B - (pkr unit mass)
LO-1 (per unit weight)
L ML 2 ~ - 2 8 - 1 L
( L ~ T - ~ (per unit
( L ( p ~ r unitweight) LT- .
h 4 - l ~ ~ (per unit mass]
M - ~ L ~ T (per unit weight)
LT-I
Use N R for a1 ternate Richness i s >1 on the rich
s i d e of the stoichiometric mixture, and <1 on the lean side; u s e K meaning rich- f
Also Appears in ASA Publicat ion
ness of the- fuel, when necessary to distinguish $ram other concepts con- noted by R
Remarks
Alternate for b/2
U s e A for cross-sectional area
Use 8 for al ternate
Use T for al ternate
Use F for other types of thrust
Z10.4,Z10.6,210. 12 I U s e e for al ternate
S e e Appendix I and Figure 1 for designation of axes
Use E for alternate;
z10.2,z10..4,z10.12 I
Use U for alternate; u s e ei ther subscripts ( e S g . , V X , V y , V z ) O r U 7 V7 W for velocity components
210.2,210.6
Z10.4,210.6,ZlO.12
See Appendix I and Figure 1 for designation of axes
See Appendix I and Figure 1 for designation of axes
Velocity, component of, along Z-axis
Weight flow, per unit time; mass-flow per unit time
LT'
MT-I $mass flow) Weight MLT-
AMERICAN STANDARD
LETTER SYMBOLS FOR PRIMARY CONCEPTS (Cod.)
Remarks
See Appendix I and Figure 1 for designation of axes
See Appendix I and Figure 1
See Appendix I andFigure 1 See h
W Defined a s arctan --; see u
Appendix I and Figures 1 and2 -
Oefined a s arcsinV V ; See Ap- pendix I and Figures 1 and 2
Use k for alternate
Angle between a horizontal plane and instantaneous direction of motion of body center of gravity, positive in climb; see Appendix I and Figure 1
Use 8 for momentum thickness of boundary layer
Differential deflections must be clearly defined; in flutter, symbol 6 i s often employed for con trol-surface angular displacement
--
Ratio of a pressure to the standard pressure of atmos- pheric air a t sea level
Also Appears in ASA Publication
210.2,Z10.3,Z10.6,Zl0.8
Z10.4,Z10.6,Z10.12 Z10.3,210.8
-
210.3, 210.8
ZlO.4,ZlO.6,ZlO. 12
Dimensions
L
L
L L
Dimensionless
T - ~ L ~ T - ~ Dimensionless
Dimensionless
Dimensionless Dimensionless
Dimensionless
Dimensionless Dimensionless Dimensionless
L~T-'
Dimensionless
L
Dimensionless
Dimensionless
Dimensionless
Dimensionless Dimensionless Dimensionless ML -IT-
Divensionless
Symbol
x
Y z z
Q
a! a! Q
P
P P
P
Y Y Y
r I-
6
6
6
6
c c c f
c
Concept
Coordinate along X-axis
Coordinate along Y-axis
Coordinate along Z-axis Deflection normal to X-Y
plane (flutter)
Angle of attack
Angular acceleration Diffusivity, thermal Nozzle-divergence half
angle
Angle of sideslip
Blade angle (propellers) Mach number relation,
I I - M ~ 1 % o r IMZ-1 1 % Nozzle-convergence half
angle
Ratio of specific heats Strain, shear Flight-path angle
Circulation; strength of a single vortex
Dihedral angle
Boundary-layer thickness
Displacement, angular, of control surface or tab
Logarithmic decrement of viscous (or equivalent viscous)damping
Relative pressure
Angle of downwash Error signal Strain, normal Turbulence-exchange
coefficient Emissivity (for radiant
heat)
LETTER SYMBOLS FOR AERONAUTICAL SCIENCES
LETTER SYMBOLS FOR PRIMARY CONCEPTS (cont.)
I Also Appears
Symbl I Concept Dimensions in ASA Publication Remarks I Efficiency Dimensionless
Dimensionless Temperature-recovery factor
Usually employed with
Symbolar i s often employed as an alternate in flutter
Angle of pitch Angle of twist of an
elastic component Momentum thickness of
boundaty layer
Dimensionless Dimensionless
L
8 1 Z10.3,210.4,Z10.6 I Use t if possible Temperature, alternate for
Temperature ratio Ratio of an absolute tempera- ture to the standard abso- lute temperature for atmos- pheric air at sea level
Dimensionless
I I I
h Mean free path L A Microscale (turbulence) L Associated with turbulence
I I 1 I decay Nozzle-divergence factor Dimensionless
Dimensionless
L
The theoretical ratio of thrust of a diverging nozzle to the thrust without diver- gence of the flow
In any given tapered wing element, the ratio of chord a t the up end to chord at the root end
Taper ratio
Wave length
Sweepback angle
Mach angle
Dimensionless
Dimensionless
Dimensionless rn M B p --, wherel i s the reference
PSI 4 m 1 length
p = - ; symbol --ii; has npc been used as an dtemate
Mass parameter of air- plane (stability)
Mass parameter of air- Dimensionless plane component (flutter)
Viscosity absolute; ML"T" Z10.2,Z10.4,Z10.12 viscosity, cnefficientof
Poisson's ratio Dimensionless z 10.3
Kinematic viscosity L ~ T " Z1O.2,21O.4,ZIO.6,210.12 v =
P
P Mass density ML" Z10.2,210.3,Z10.4,210.6, Such h a t pl/P = pressure Z10.8,Z10.12 2
Angle of sidewash Relative density
Dimensionless Dimensionless
Dimensionless
Ratio of a density to the standard density of aanos- pheric air at sea level
Solidity (propellers)
u Stress, normal; stress ML"T'~ 210.2,210.3 (intensity)
T
7 Stress, shearing M L ' ' ~ ~ 210.2,210.3 7 Time, alternate for T Z10."4,210.6,Z1O.12 Use t (Ic) if possible 7 Time ratio Dimensionless
AMERICAN STANDARD
LETTER SYMBOLS FOR PRIMARY CONCEPTS (Cont.)
Remarks
See Appendix I and Figure 1
See Appendix I and Figure 1
Use a a s alternate for angular velocity
Also Appears in ASA Publication
210.2,210.3,Z10.4,210.6 210.8,210.12
Dimensions
Dimensionless Dependent on
definition Dimensionless
Dimensionless
Dependent on definition
T'
Symbol
9 9 9
9
+ I
o
If possible, use o for I angular velocity
Concept
Angle of 1011 (or bank) Potential function
Helical angle of advance (propellers)
Angle of yaw; effectiw helix angle
Stream function
Angular velocity; circu- lar frequency
T" 8 Angular velocity, a l t e r nate for; mrticity
Letter Symbols for Secondary Concepts SYMBOLS FOR USE AS SUBSCRIPTS
(Note: Letter symbols for primary concepts may also be used for subscr ip ts ; s e e Foreulord)
--- ---
Subscript Symbols Concept Remarks
a Absolute Added; additional
I Adiabatic See also ad for alter-
I nate; u se aw for adiabatic wall
Ai 1 eron Air; relative to air Allowable Ambient See am for alternate Available
ac Aerodynamic center (of a surface or airfoil section)
ad Adiabatic, alternate for
Subscript 1 s y ~ b o h ~ Concept 1 Remarks
Damping, general When possible, II I
- -
I replace bv subscript 11 1 1 indicating directio;
Diffuser, duct d ischarge
Dry Drag
1 of motion-or of angular velocity causing the damping
Use only for the dis- charge coefficients of nozzles and orifices
Use d with section coefficient
a m Ambient, 1 1 e Earth bound As applied to axes alternate for i or to vector
av Average See also m for mean A Axial components; s e e
See A~pend ix I / Appendix I1 b Base Effective Use eff a s alternate
Basic Elevator B ending See h for alternate Endurance Blade Engine Burner; burnt; Equivalent 1
I - burning Exhaust; exhaust , I t i c I 1 nozzle; exi t see also i for fuselage e ff ' Effective, alternate I for
Calibrated Use cal for alternate ELL / Euler (number) I I ____-__- Chord -- -A
I / -_ _ - --- -
corr C P C r
Climb Combustion; com-
bustion chamber
I
I Compressibility,
I I f Use b for combustion I chamber in gas- 1 turbine power plants to distinguish from c for compressor
Fin Flap: flipper Fluid; l i a ~ i d phase Flutter Friction, friction
factor - I! i Fuel
- ---- - -. - -- - - -- -. -- - -- Set I for alternate
- - - -
compressible, -- / Fuselage; body / See also 4 for body compression, - I
1; ~ compressor Force; thrust ; See also 7' for thrust Coolant Use c l for alternate Frontal -
Critical Use c r for alternate /;r I Froude (nunber) 1 - - - - - . .
Calibr ated, alternate for
Calculated Center of gravity Coolant,
alternate for Corrected Center of pressure Critical,
alternate for Cross wind Cauchy (number)
See Appendix I
Gage Gas; combustion
gases ; exhaust gas Gross
Grashof (number) Graetz (number)
May require care in definition
Bending, alternate fo -- - - --
Heat; heat exchanger -- - - - --
Hinge 1 -
AMERICAN STANDARD
Remarks
May require care in definition
See also N
See Appendix I
-
See Appendix I and Figure 2
See o x for alternate
See superscript aster- isk (*) for charac-
value; s e e sl for teristic or reference
alternate for sea- level conditions
See para for alternate
See also for moment
May require care in I definition 1
LETTER SYMBOLS FOR SECONDARY CONCEPTS (Cont.) SYMBOLS FOR USE AS SUBSCRIPTS
Subscript Symbols
n
nP
N
Nu
o
Remarks
See also r for root Used primarily a s
applied to heat of combus tion
As applied to axes
Subscript Symbols
h
H
i
Concept
Net
Normal to a surface Nozzle Yawing moment
Neutral point (of a configuration)
Normal Normal, perpendic-
ular to X-Y plane Nusselt (number)
Initial Outer Oxidizer Profi le (drag) Standard or refer-
ence condition;
R
Concept
Horizontal Hub Higher
Ideal Incidence (angle) Indicated; observed Induced Inertial
min M Ma
Reference condition, alternate for
Relative Root Rotor; rotational Rudder
Resultant Right
conditions standard sea-level
Zero lift Oxidizer, alternate
for
Paras i te ; parasitic Polar Potential Pressure, based on Propellant; propel-
1 er; propul sive; tailpipe
Paras i te , alternate for
Power P ec le t (number) P randtl (number)
Torque (propellers and0rotary-wing aircraft)
--- Radial Ram
Recovery
1 Inlet; intake; input Inner; internal Interference
S e e l? for alternate
I O X
P
para
P P e P r
Q
r
i
k
1
lam lat le L
m
max
or to vector compo- nents; s ee Appendix I
M inimum Moment, in general
i Mach (number) When used as a sub- s cript,thi s double- 1 etter form i s recommended for consistency with designations for other dimension- l e s s numbers
J e t
Kinetic
Liquid phase,
/ alternate for Local condition,
just outside boundary layer
Lower (surface) Rolling moment Section l if t
Laminar Lateral Leading edge Landing conditions Lef t Level Lift
Lower
Mean
Pitching moment
Maximum
May require care in definition
Recommend use of
I i tal icized symbols, and script ell (lower case) if italicized symbol i s not available
See Appendix I
Use 1 with section coefficient
Use primarily a s a p plied to heat of combustion
See a lso av for average
See Appendix I
-- Subscript Symbols
R
tab
turb T
LETTER SYMBOLS FOR AERONAUTICAL SCIENCES
LETTER SYMBOLS FOR SECONDARY CONCEPTS (Cont.)
A -- - -
Remarks
------& See s for alternate
--
--
Positive in positive X direction; see Appendix I
Positive in positive Y direction; see Appendix I
-- . - --
Concept
Ultimate Upper (surface)
Vertical Vapor Volumetric
Wall; surface Wind \king U'eber (nuvber) Wave
Excess Component parallel
to X-axis
--- Yield Component parallel
to Y-axis
---
AS SUBSCRIPTS --
Subscript Symbols
u
-
v
vol
U)
we W
x s X
Y Y
Concept
Rotor, alterwte for; rotational, alter- nate for
Reynolds (number)
Shaft Shear Slipstream Speed-power Stability
Stabilizer Surface, alternate
for Standard sea-level,
alternate for Specific Static Standard, alternate
for Stall conditions Stanton (number)
SYMBOLS FOR USE
Remarks
-
As applied to axes o r to vector com- ponents; see Appendix I
z
6
6
8
P
a
0,1,2,3,4,.
m
Tab Tail Tangential Tensile Throat Tip Total; isentropic
stagnation con- ditions
Turbine
Tab, alternate for Trailing edge Theoretical Throat, alternate for Turbulent Take-off and termi-
nal velocity conditions
Thrust (for propel- lers and rotary- wing aircraft
Tunnel
Component parallel to Z-axis
Angular deflection of a control sur- face or tab
Boundary-lay er thickness, based on ---
Elastic angular deflection; twist --
Density, based on
Condition which ap- plies immediately after shock
Station subscripts
Undisturbed; free- stream
See Tab for alternate
See th for alternate
See also 'superscript zero ( O )
-
-
-
*
See also F for force; thrust
Positive in positive Z direction; see Appendix I
--- -
-
-
AMERICAN STANDARD
LETTER SYMBOLS FOR SECONDARY CONCEPTS (Cont. )
SYMBOLS FOR USE AS SUPERSCRIPTS
Super- script
Symbol Concept Re marks
'(Dot)
"(Double- dot
'(Prime)
'(Prime)
"(Double prime)
- (Bar)
+ (Plus)
(7.ero)
* (Asterisk)
First derivative, with respect to time
Second derivative, with respect to time
First derivative, with respect to distance; effec-
These symbols are written o-rer, not after, the principal symbol to which they are applied
Used in turbulent-flow analysis to identify a turbulent-veloci ty component, e.g., - u ' = u - U ; also used to designate the pressure measured by a total-head tube
To designate a
in a supersonic , stream, e.g., p
Alternate for sub- second s e t of axes, or quan- t i t ies related
scripts
to such a x e s Second derivative,
with respect to distance
Mean value
A dimensionless measure of the quantity con- sidered
Total; isentropic stagnation con- di tion s
Characteristic or reference value; critical condi- tions at (M = 1
Commonly used for value per unit area
This symbol i s written over, not after, h e principal symbol to which i t i s applied
Commonly used for boundary-layer flow and for flow in pipes
See also subscript t
A s applied to u for friction velocity (u*); as applied to 6 - displace- ment thickness of boundary layer (6*)
Appendix I
RECOMMENDED DESIGNATIONS F O R SYSTEMS O F AXES, ANGULAR RELATIONSHIPS, AND QUANTITIES R E L A T E D T H E R E T O
1 1. Axes:
X - Longitudinal body axis, in vertical plane of symmetry and preferably coincident with centerline of symrnPtrical body or parallel to thrust line of asymmetrical fuselage; positive forward' .
Y - Lateral body axis, perpendicular to vet- tical plane of symmetry; positive to right when looking forward (Same a s lateral s tab~l i ty axis)'
Z - Vertical body axis, in vertical plane of symmetry and perpendicular to the longi- tudinal and lateral body axes; positive downward'
xw - Longitudinal wind axis, parallel to the relative wind; posltive forward','
Yw - Lateral wind axis, perpendicular to the longitudinal and vertical wind axes positive to right when looking forward"'
zw - Vertical wind .axis, in vertical plane of
symmetry and perpendicular to the rela- tive wind, positive downward (Same a s vertical stability axis)'"
xs - Longitudinal stability axis, parallel to the projection of the relatlve wind on the vertical plane of symmtry,positive forward
Ys - Lateral stability axis (same a s lateral body axis)
Zs - Venical stability axis (same a s vertical wind axis)
Xc - Longitudinal earth axis, lying in plane tangent t o earth's average surface or in plane perpendicular to the direction of gravitational force"4
Ye - Lateral earth axis, lying in plane tangent to earth's average surface and perpen- dicqly to the longi tdinal earth axis; positive to right when looking along positive direct Ion of longitudinal earth axis'
Ze - Vertical earth axis, mutually perpendic- ular t o longitudinal and lateral earth axes, positive downward'
Xi - Longitudinal inertial axis, usually as- sumed to l ie initially in plane tangent to earth's average surface, but fixed with relation t o space"4
Yi - Lateral inertial axis, usually assumed to lie initially in plane tangent to earth's average surface and perpendicu- lar to the longitudinal inertla1 ax+ but fixed with relation to space; posltlve to right when looking along pos~t ive direc- tion of longitudinal inertial axis'
Zi - Vertical inertial axis, mutually perpen- dicular to longitudinal and lateral iner- tial axes; positive downward initially'
2. Angles:
,, 0 - Angles of attack and sideslip, respective- l y , a s defined in Letter Symbols for Primary Concepts (page 12); a lso see Figures 1 and 2.
y - Flight-path a n le, a s defined in Letter Symbols tor ffrlrnary Concepts (page 12).
$, 6,+- Angles of yaw, pitch, and roll, respec- tlvely; a system of three angles which uniquely define with reference to one coordinate s ys tem (e.g., earth axes), the orientation of a second coordinate system (e.g., body axes). Any orienta- tion of the second system is obtainable from that of the first by rotation through each of the three angles in turn,the sequence of which is important. The preferred order is !l!usuated in Figure 1, where the posltive directions are defined. Any deviation from this order must be clearly indicated.
'nody axes are orthogonal axes fixed in the body, Choice of origin and orientation depend on nature of problem and should be clearly defined (See Figures 1 and 2.)
2 ~ i n d axes are orthogonal axes fixed with respect to the instantaneous relative wind. Origin i s at the center of gravity or designated moment reference point.
3 ~ f only one set of a3es i s involved, the subscript may be omitted provided the e x e s ere clearly defined. If desired, for simplicity, a prime ( ) may be used a s an alternate for the subscript provided it8 significance i s clearly indicated.
'~ irec t ions of longitudinal earth and inertial axes must be clearly defined.
APPENDIX I
1 Ezample: Let the orientation an les ($, ?, 4) be zero
when the aircraft axes ate afigned wlth the earth axes (i.e., straight and level flight along the posi- tive direction of t h X - axis). Any other orientation of the aircraft is then %ached by:
Rotation (1). Yaw about Z-axis, through angle $, until X-axis l ies in the vertical plane con- taining i t s ultimate position:
Rota ion (2). Pitch about Y-axas, through angle b, until X-axis points in its final direction. Rotation (3). Roll about X-axls, through angle 4,
until aircraft has the required orientation. 3 . Angular Velocities:
p, q, r - angular velocities of bodpaxis system (X, Y , 2, respectively), p-ositive clock- wise w k n looking in pos~tive direction of axes
4. Linear Velocities: u, v , w - components of resultant velocity V
along body axes (X, Y, Z, respectively)
5. Forces and Force Coefficients:
Name Positive - Force Coefficient Directim
BODY AXES: axial FA f-Fx) C A ( z x ) -X side F~ Y Y
WIND AXES: h a g F~ C~ 'Xw crosswind F~ C~ w
I lift FL C~ -zw STABILITY AXES:
drag (approx.) Ff) Co' -Xs side C~ Y,fY) lift F~ L -Zsf-2 J
6. Moments and Moment Coefficients:
Name Moment Coefficient Positive Direction
ABOUT BODY AXES: rolling M~ c 1 is clockwise look-
ing forward
pitching yawing
MY m raises nose
C n rotates nose to right
ABOUT WIND AXES: rolling M ~ w is clockwise look-
C l * ~ ing forward
pitching 'm, w raises nose
yawing M ~ ~ v 'n, w rotates nose to right
ABOUT STABILITY AXES: rolling M ~ s C1, i s clockwise look-
ing forward
pitching /My) 'rn raises nose yawing M ~ s f"zw) 'n,w rotates right nose to
NOTES:
1.) Symbols in parentheses may be used a s alternates.
2.) Force coefficient i s d e f i ~ e d a s force d i v i d ~ d by the product of two quantities, namely, the dynamic pressure (g) and a reference area.The choice of reference area depends on the nature of the problem, and the particular reference area del,ends on the nature of the problem, and the particular reference area must be clearly indicated.
3.) Moment coefficient i s defined a s mon~ent divided by tt,e product of three quantities, namely the dynamic pressure (9). a reference area, arid a reference length. The choices of reference area and reference length depend on the nature of the problem. The reference arec. must be the some a s that used for the force coefficients. The reference length need not be the same for p-itching, yawing and rolling moments and, along with the respective moment reference points, m u s t be clearly indicated for each:
APPENDIX I
Xe Figure I.- Orientation of coordinate systems, showing direction and sense of Euler angles.
NOTE:
P O S I T I V E SENSE OF AXES, VELOCIT IES, AND ANGLES I S l NDl CATED BY ARROWS.
Appendix II
ALPHABETICAL INDEX BY CONCEPTS 1
Sub- Super- script script
Symbols Symbols 1 Concept 'rincipal Symbols
SU b- script
Symbols Principal Symbols Concept
Angular velocity of body-axis system about X-axis
Angular velocity of body-axis system about Y-axis
Angular velocity of body-axis system about Z-axis
Absolute Absolute temperature Acceleration, angular Acceleration due to
gravity Added; additional
Adiabatic
Adiabatic wall Advance, helical angle
of (propellers) Advance ratio
(propellers) Aerodynamic center Aileron Air; relative to air Airplane Allowable
Area, cross-sectional Area moment of inertia Area, surface or projected Aspect ratio
Attack, angle of, relative to wind
Avail able Average Axial Bank, angle of Barometric Base Basic Bending
a av
- bar b b
1: b - - -
I7 -
A1 ti tude
Ambient
Angle of attack, Angle of bank Angle of blade
(propellers) Angle, dihedral Angle of downwash Angle, effective helix Angle, flight-path ,
Angle, helical, of advance (propellers)
Angle of incidence Angle, Mach Angle of pitch Angle of roll Angle of sideslip Angle of sidewash Angle of sweepback Angle of twist of an
elastic component Angle of yaw Angular acceleration Angular displacement
of control surface or tab
Blade Blade angle
(prope 11 ers ) Blade thickness Blade width
(propellers)
Boundary layer, dis- placement thickness of
I Boundary layer, momentum thickness of
Boundary-layer thicknes: , Burner; burnt; burning I
6 b
calc
c:1 Ca ac cg C P
C
Calculated
Calibrated
Cauchy (number) Center, aerodynamic Center of gravity Center of pressure Chamber, combustion Characteristic or
reference value
Angular momentum h g u l ar velocity;
circular frequency
APPENDIX I1
I APPENDIX I1
I Note: Axis designations for particular coordinate systems may 'be used to connote direction (e.g. . F X , F Y , Fzl.
Super- script
Symbols
- -
- - - - - -
- -
- - - - -
-
- - -
- -
-
-
-
- - - - -
- - - - - -
Principal Symbols
1 i k
- u*
N ~ r
-
R c
N~ z
N~ r g
- U k r
- i,
(2 4
C'
d
4 -
1 i
1 I
Concept
Frequency, circular; angular velocity
Frequency, reduced (flutter)
Friction; friction factor Friction velocity Frontal Froude (number) Fuel Fuselage; body
Gage Gas; combustion gases ;
exhaust gas Gas constant Gas velocity (rockets) Graetz (number) Grashof (number) Gravity, accelerat ion
due to Gravity, center of Gross Gust velocity Gyration, radius of Gyration, reduced
radius of (flutter)
Heat; heat exchanger Heat-flow rate, per unit
area, per degree, across a boundary surface
Heat, quantity of Heat, specif ic quantity
of; quantity vf heat , per unit mass or per unit weight
Heat transfer coeffi- cient , over-all
Helical angle of ad- vance (propellers)
Helix angle, effective Higher Hinge Hori ton t a1 Hub
Ideal Impulse Incidence (angle) Indicated: observed Induced Inertia, area moment of Inertia, mass moment of Inertial Initial
Sub- script
Symbols
-
-
- f F
Fr f f
g g
G z Gr
Cg g
h -
H h h h
1
i i i
--- i o
Super- scr ip t
Symbols
- - - - - - -
- - - - - -
-
- -
-
- - - . - -
- -
-
- -
- - - - - - - - -
-
Sub- script
Symbols
- - -
8
-
e -
e
: e -
-
- -
-
e - Eu xs
h e
g e
-
f f f -
i f
F See
footnote -
N (D
-
Concept
Effect ive helix angle Effectiveness, span Efficiency Elas t ic angular deflec-
tion; twist Elasticity, modulus of;
Young's modulus Elevator Emissivity (for radiant
heat) Endurance Energy
Energy, internal
Energy, internal, per unit mass or per unit weight
Engine Enthal~y, ner unit mass
or per unit weight; specific enthalpy
Enthalpy; total hea t con t en t
Entropy Entropy, per unit mass
or per unit weight; specific entropy
Equivalence ratio; richness(combustion)
Equivalent Error Signal Euler (number) Excess Exchange, he at Exhaust; exhaust
nozzle Fxhaust gas Exit
Factor, for comparison purposes
Factor, friction Fin Flap Flight-path angle Flipper Fluid; liquid phase Flutter Force: resultant force
Force cornponen t s
Force, cross-wind Force, normal Free-stream;
undisturbed Frequency, cyclic
Principal Symbols
6 e 'I -
E
- a
-
[ i -
h
fi
s s
6 c - - - -
- -
k'
-
- Y - - -
F -
c f l -
f
APPENDIX I1
Concept
Inlet Inner Input Intake Interference Internal
Internal energy
TnternaI energy, per clnit mass or per unit weight
Isentropi c stagnation conditions; total
Jet Joule's constant;
rnechan ical equivalent of heat
Kinemaric viscosity Kinetic
Laminar Landing (conditions) Lateral Leading edge Left Length; distance Length, correlation
(turbulence) Level Lift; lift, per unit
length (flutter) Lift curve, slope of Lift, section Lik, zero Liquid phase; fluid Load factor Local condition, just
outside boundary layer
Logarithmic decrement of viscous (or equi- valent viscous) damping
Lower Lower (surface)
Mach angle Mach number
Mach number relation
1-,Va ' a M a - 1 % Mass; mass, per unit
length (flutter)
S u p ~ r - scnp t
Symbols
- - - - - - -
-
-
O (zero)
- -
-
-- -
-
-
-
- - - -
-
-
- -
-
-
Principal Symbols
{: 1: - -
- 1
v -
- - - - -
1 L -
L
a - - -
n -
8
- -
n; {c.
P
m
Sub- script
Symbols
I
i i i I
i -
-
t - i -
- k
lam I,
lat le L - -
L C
- 1 o I -
1
-
L 1
- Ma
-
-
Concept
Mas's density Mass-flow, per unit
cross- sec t~ona l area, per unit time; mass velocity
Mass-flow , per unit time4 weight flow, per unit time
Mass moment of inertia; mass moment of iner- tia, per unit length
Mass parameter of alr- plane (stability);
Mass parameter of air- plane component (flutter)
Mass velocity; mass- flow, per unit cross- sectional area, per unit time
Maximum Me an Mean free path Mean velocity compo-
nent (turbulent flow) Measure (dimensionless)
of the quantity con- sidered (boundary- layer flow; flow in pipes)
Mechanical equivalent of heat; Joule's constant
Microscale, turbulence Minimum Modulus of elasticity;
Young's modulus Modulus, shear Moment; torque Moment per unit
length (flutter) Moment of inertia, area Moment of inertia, mass;
mass moment of i n e ~ tia, per unit length
Moment, pitching Moment, rolling Moment, yawing Momentum, angular Momentum thickness of
boundary layer
Net Neutral point Normal
Principal Symbols
P G
w
I
-
CL
G
- - X -
' -
1
X - E
G M M
1 I
- - - H 8
- - N
Sub- script
Symbols - -
-
-
- - -
max m - -
-
-
- min - - - -
- -
m 1 n - -
n nP -
Super- script
Symbols
- -
-
-
- - ---
- - (bar) - - (bar)
+ (plus)
- - - -
- - -
- -
- - - - -
- - -
APPENDIX I1
Sub- script
Symbols
P CP - - d -
P o -
P P P - - -
r - - -
r - - -
r - - -
- i; r a - -
I2 - - -
Principal Symbols
P - i P -
- - S
- - -
Q 9
4
- r k r
- R
I 1 r
- Tr
k
r
-
-
- -
o 6 - F
n 1 'U
Super- script
Symbols
- - - - -
'(prime)
- - -
- - - - -
-
- - - - - - - - - -
-
- -
* (asterisk)
- - - - - - - -
Concept
Pressure Pressure, center of Pressure, dynamic Pressure, relative Pressure, static Pressure, total-head
(measured by total- head tube)
Profile Profile (drag) Projected area; surface
area Propellant Propeller Propulsive
Quantity of heat Quantity of he at, per
unit tirn e Quantity of heat, per.
unit mass or per unit weight; spec~fic quantity of heat
Radial Radius Radius of gyration Radius, reduced, of
Ram gyration (flutter)
Ran ge Ratio of specific heats
Ratio, time Recovery Recovery factor,
temperature Reduced frequency
(flutter) Reduced radius of
gyration (flutter) Reference (or standard)
con di ti on s Reference (or charac
teristic) value Relative Relative to air Relative density Relative pressure Resultant Resultant force Resultant velocity
Revolutions, per unit time
Concept
Normal, perpendicular to X-Y plane
Normal (perpendicular to a surface)
Normal force Normal strain Normal stress; stress
(intensity) Nozzle Nozzle, exhaust Nozzle-convergence
half angle Nozzlcdivergence
factor NO zzl e-divergence half angle
Number, in general Numbers:
Cauchy Euler Froude Graetz Grashof
Mach
Nussel t Peclet Prandtl
Reynolds
Stanton Weber
Nussel t (number)
Observed; indicated Outer
Oxidizer
Parasite; parasitic
Peclet (number)
Phase, liquid
Pitch, angle of Pitching moment Poisson's ratio Polar Potential Potential function Power Prandtl number Precise or effective
value
Sub- script
Symbols
N
n
N - -
n e -
- - - Ca Eu Fr Gz Gr
,Ma
Nu Pe P t
Re
S t We
Nu
i o
{ o z
P
{P;;
( f . - - -
P P
P - P r
-
Principal Symbols
- -
N c (I
- -
0
h
a
N
N gu Pr 4:
{ to Pu
{ $: N": G e
N~ u
- - -
-
'P e -
8 m CL - - $' "'P I
-
S F ' scrlpt
Symbols
- -
- - - - - - - a
- - - - - - -
- -
- -
, -
- - - - -
- - - - - - - - - - -
'(prime)
Sub- script
Symbols
-
- s s
s t
S
1 st:
/: st -
0,1,2,3,4.. St
- - - - -
-
- -
W
S - -
-
- - - - - - - - -
t T
th -
Principal Symbols
v
V - - - - - -
- -
P - "'s t
J
c
5 F
T
T
g
.t- - S
A
L b t
P T t h T
1 f!i 8
q r - -
- k
Super- script
Symbols
-
- - - - -
O (zero) - -
- - - - - - - - - -
- - - - - - -
-
- - - - - - - - - -
- -
II
Concept
Specific volume; volume per unit mass or per unit weight
Speed Speed-power Stability (see Appendix
n) Stabilizer Stagnation conditions,
isentropic Stall conditions Standard (or reference)
conditions
Standard sea-level conditions
S t a t ~ c Static pressure Station subscripts Stanton number
Stiffness constant, torsion a1
Strain, normal Strain, shear Stream function Stream thrust; thrust
(combustion) Stress, normal; stress
(intensity) Srress, shearing Structural damping
coefficient
Surface
Surface area; projected area
Sweepback angle
Tab
Tail Tailpipe Take-off conditions Tangen ti a1 Taper ratio Temperature, absolute
Temperature, general
Temperature ratio Temperature-recove ry
factor Tensile Terminal-velocity
conditions Theoretical Thermal conductivity
APPENDIX
Super- script
Symbols
-
-
- - - - - -
- - - -
- - - - - -- -
- - - - - - - - - -
-
--
-
- -
-
Sub- script
Symbols
Re
-
R - 1 r
{R' r
- -
1 -
- s S
- - -
o
- - s - - - - -
S P -
-
-
-
- -
-
Concept
Reynolds number
Rithness; equivalence ratio (combustion)
Right Roll, an#e of Rolling moment Root
Rotor; rotational
Rudder
Scale, turbulence 'Section coefficient 9ection lift Semichord, alternate for
c/2 (flutter) Semispan, alternate for Shaft Shear Shear modulus Shear strain Shearing stress Shock, condition which
applies immediately after
Sideslip, angle of Sidewash, angle of Slipstream Slope of lift curve Solidity (propellers)
Sound, velocity of
span Span effectiveness Specific Specific enthalpy;
enthalpy per unit weight
Specific entropy; entropy per unit mass or per unit weight
Specific heat, at con- stant pressure
Specific heat,at con- stant volume
Specific heats, ratio of
Specific internal energy; internal energy, per unit weight
Specific quantity of heat; quantity of heat, per unit mass or per unit weight
Principal Symbols
{ 4e (4 - + - - -
-
L c
-- b
s - - G Y T
-
I3 u - a u
/: b e - h
s
c P
cr
I r u
4
Super- script
Sypbols
-
-
-
-
- (bar)
'(prime)
-
--
- - - - -
- - - - -
- - -
7
- - - -
- - - -
Note: Axis designations for particular coordinate systems may be used to connote direction (e.g., V x , V y , Vz).
Principal Symbols
P
4
r
U y v , W
- -
u
C'
w
u* c U C
- 1: w
- P
P
s" V 2'
v - r !2 - -
A
I1
Concept
Velocity, angular, of body-axis system about X-axis
Velociw, angular, of body-axis system about Y-axis
Velocity, angular, of body-axis system about Z-axis
Velocity components
Velocity (mean) com- ponent (turbulent flow:
Velocity (turbulent) component, particular
Velocity, component o f , along X-axis
Velocity, component of, along Y-axis
Velocity, component of, along Z-axis
Velocity, friction Velocity, gas (rockets) Velocity, gust Velocity, mass; mass-
flow, per unit cross- sectional area, per unit time
Velocity of sound
Velocity (conditions), terminal
Velocity vector, magni- tude of
Vertical Viscosity, absolute;
viscosity, coefficient of
Viscosity, coefficient ofi absolute viscosity
Viscosity, kinematic Viscous damping, loga-
rithmic decrement of Volume Volume, per unit mass
or per unit weight; specific volume
Volumetric Vortex, strength of a
single; circulation Vorticity Wall; surface Wave Wavelength
Sub- script
Symbols
-
-
-
See footnote
-
-
- -
- - - - -
- T
- v -
- - - - -
- vol -
- w B -
APPENDIX
Super- script .
Symbols - - -
-
-
- -
-
- - - -
-
O (zero)
'(prime)
-
- - - -
- - - -
'@rime)
-
- -
-
- -
Sub- script
Symbols
- -
8
-
1 t i F
T
- -
t M Q
-
t
-
-
t e T
1 -
- - -
twb -
H
u m
u
v -
Concept
Thermal diffusivity Thickness Thickness, blade
(propellers) Thickness of boundary
1 ay er Thickness (momentum)
of boundary layer
Throat
Thrust; stream thrust (combustion)
Thrust (propellers and rot ary-wing aircraft)
Time
Time ratio Tip Torque; moment Torque (propellers and
rotary- wing aircraft) Torsional stiffness
constant Tota1;isentropicstag
nation conditions Total-head pressure
(measured by total- head tube)
Total heat content; enthalpy
Trailing edge Tunne 1 Iurbine Turbulence correlation
coefficient Turbulence exchange
coefficient Turbulence mi croscale Turbulence scale Turbulent Turbulent-velocity
component, particular Twist of an elastic
component, angle of
Ultimate Undisturbed; free-
stream Upper (surface)
vapor
Velocity, resultant
Velocity, angular; circular frequency
Principal Symbols
(Y
t h
2
8
- F
T
1: T -
M Q
1 --
-
H
- - -
R
45
X L -
0
- - -
p w R
APPENDIX I1
American Standards for Abbreviations.
Symbols and Charts 'I
TITLE OF STANDARD
Let te r Symbols for Aeronautical S c i e n c e s ( Y 10.7.1954) ................... .... ............................. Abbreviations for Scient i f ic and Engineering Terms ( Z 10 . 1- 194 1) .................................... L e t t e r Symbols for I ~ y d r a u l i c s ( 2 1 0 . 2- 1942) .......................................................................... Let te r Symbols for l l echanics of Solid Bodies (210.3- 1948) ...............................................
........... Let te r Symbols for Heat and Thermodynamics, Including Heat F l o w (210.4- 1943) Let te r Symbols fnr P h y s i c s (210.6. 1948) ......................................................................... Let te r Syrribols for Structural Analys i s (Z10.8- 1949) .......................................................... Let te r S y r b o l s for Radio ( Y 10.9. 1953) ............................................................................. Let te r Symbols for Meteorology (Y 10 . 10- 1953) .................................................................. Let te r Symbols for Acous t ics (Y1O.ll-1953) ................................................................ Let te r Symbols for Gear Engineering (86.5- 1954) .......................................................... Drawings and Drafting Room P r a c t i c e (Z14 . 1- 1946) ...................................................... Graphical Symbols for
. :Y'elding and Instruct ions for The i r U s e ( Z 3 2 . 2 1- 1949) ........................................ Plumbing (232.2.2-1949) . . . . . . . . . . . . . . . . . . : .......................................................... P i p e F i t t ings , Valves, and Piping (232.2.3-1949) ...................................................... FIeating, Ventilating, and Air Conditioning (71.32.2.4- 1949) ............................................. Railroad U s e ( 7 3 2 . 2.5- 1950) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ..
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ileat-Power Apparatus (232.2.6- 1950) ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abbreviations for U s e on Drawings (232.13-1950)
Engineering and Scient i f ic Char t s for Lantern S l ides (213.1- 1932; reaffirmed 1947) . .
. . . . . Fngineering and Scient i f ic G r a ~ h s for Publ ica t ions (Z15.3- 1943;reaffirmed 1947) . . . . . . . . . . . . . . . . . . . . Time Ser ies Char t s (215.2-1938; reaffirmed 1947)
PRICE . 81.25 0.50 0.50 0.50 0.65 1.00 0.50 1.00 1.00 1-00 1.00 1.50
20% Discount to ASllE r.rernbers .
A binder is available for holding these standards . It holds ttc~enty-five (25) pamphlets and gives every advan. tage of a bound booh v i t h tlze at!(l'ed convenience zuhich comes from t h e ability t o insert. remove. or transpose s ec t ior~s o f the contents . Price $3.25 postpairl .
A corirplete l i s t of Alr~erican Stanrlarrlr pub l i s l~er~ by I'Ae American Society o f : lechanicd bz7r~gineers obtairlable upon request .