TM 1-320
8 AIR CORPS
cross section. K for a concave hemisphere is about 0.00389 (see
fig. 7).
( 4) Oonvero hemisphere.- For a hemisphere with the convex side
facing the direction of motion or pointing against the wind the
X= . 00389
:
-
Jl'louam 7 .-Air stream tlowing by a concave hemisphere.
x = .oooea
FIGURE ~.-Ai r stream flowing by a convex hemisp here.
X= .0008
:
:
FIGURE 9.-Air stream flowing by a sphere.
resistance is much less than for the concave hemisphere shown in
figure 7. The resistance of the convex hemisphere is much less than
t.hat of a flat plat of the same cross section or exposed area. Th~
coefficient of resistance is found to be about 0.00082 (see fig. 8}.
AIRSHIP AERODYNAMICS
TM 1-320
(5) Sphe-1-e.-The air flow around a sphere (which more closely ap
proaches a streamline form) . is shown diagrammatically in figur e 9.
It will be observed that the spreading out of the lines of flow before
reaching the sphere is less marked than for the flat plate in figure 5.
The coefficient of resistanc e of a sphere va.ries somewhat with the speed,
R.= 1.00 fat- flat -pla.te
R.= .83 where L:o. = t
R= .77 where L: 0=3:1
FIGOREl 10.-Cyll nders.
but for ordinary velocities its value is about 0.008. The sphere is the
simplest geometrical form and is the most efficient shape for maximum
volume per unit weight but has a greater resistance than the more
perf ect streamline form (see fig. 9).
( 6) Oylinder ( longitudiMl aaJis horizontal) .- The resistance of such
cylinders decreases wl.th length until the fineness ratio is approxi -
X= ·?0123 fort= .5
,.
,. , ...
li'I GOR!l 11.-Alr stream flowing by a cylinder (arts normal to air t'low).
mately 4 to 1, after which it increases. The increase is due to the effect
of slcin friction which will be discussed later. The relative resistance
of cylinders as compared to that of a flat plate of the same cross sec
tion is as shown in figure 10. Where the fineness ratio is 4 to 1,
K = 0.00205.
TM 1-320
8 AIR CORPS
(7) OyUnderr (vertical).-When a cylind er of given cross-sectional
area is placed with its axis of revolut ion at right angles to the direction
of motion the resistance depends upon the fineness ratio of the cylinder.
When the length and diameter of the cylinder are the same the coeffi
cient of resista nce is only slig htly greater than for a sphere of the same
L K = . 0006 for D = 4
--
FIGURE 12.-Air stream flowi ng by a cyli nder (hemisph erical ends) .
.
cross-sectiona l area. When the length-diameter ratio is increa sed to
4 to 1 the coefficient of resistance is approximate ly doubled, or
K =0.0018, and if the length-diamete r ratio is reduced to one-half (or
0.5/ 1) t he coefficient of resistance is increased 27 percent, or K =0.00123
(see fig. 11). .
(8) Cylinder with hemispherical ends.-It is possible to reduce
greatly the resistance of a cylinder by capping the ends with h emi-
WIRJ:S CABLES
X: . 0026 K: . 0013
-~
K =. 0015 X = • 0029
·~ •
FIGURE 13.
spheres . The resista nce is reduced to 20 percent of that of a cylind er
with flat ends. The value of K for a cylinde r with hemispherica l ends
·and a fineness ratio of 4 is approximatel y 0.0006 (see fig 12).
(9) Wires and cables.- Wires and cables may be considered as cylin
ders of very long length . E xperiments show that the resista nce of wire
or stranded cable when placed norma l to direction of motion is very
nearly equal to the resistance of a flat plate of the same projected area.
The gain by the circular form of the wire is counter balanced by its very
Affi SHIP AERODYNAMICS
TM 1-320
great length. The resistance of a long, narrow object perpendicular to
direction of motion is greater than that of a more symmetrical form.
The experimentally determined value of the coefficient of resistance is
0.0029 for stranded cables and 0.0026 for smooth wires. K is almost in
N.PL. "I
N.P! .. 12
. Fimnessl!t:dto 4j/!. K=.ooo4
N.PL 11'..3
'
/'inmes.s ~crlto 41% K:.OOQ38
NP.J.. ..-4
~-r-:- ·- ·- · ~
' "/ /
-~ /
l'lnen~ss Raho 2/1 1<Q.ooo8:J
FIGURE 14.-Struts .
dependent of the diameter for all sizes of
wire and cable. Stranded wire or cable ha~
a resistance about 14 percent greater than
solid wire.
(a) The above discussion relates only to
wires and cables perpendicular to the wind
direction or direction of motion. . When a
wire or cable is inclined to the perpendicula r
its resistance is very much decreased as the
air flows around it in more uniform stream
lines or in a more gradual curved path. An
inclination of about 30° from the vertica l
reduces the resistance 20 percent and an in
clination of · 45° reduces the resistance 50
percent.
(b) When two wires or cables are close
together and placed one just behind . the other
there is a reduction in resistance due to
shielding of the second wire by the first. If
they are placed very close together their
combined resistance is considerably less than
the resistance of one wire .alone, as the two
wires have the effect of an increased fineness
ratio. If they are spaced more than 3ljz di
ameters their combined resistance becomes
greater than a single wire but is still less
than the resistance of the two wires tested
separately. This shows that if two wires or
cables are close together (within 5 diameters
of each other) it is very advisable to put a filler block in between
them, thus preventing the air from flowing in between them and .
giving them the advantage of a single member of high fineness-ratio.
If the two wires are streamlined in this way their combined resistance
can be kept down to about 50 percent of the resistance of a single
wire until their fineness ratio becomes greater than seven. The high
value of the resistance caused by wires and cables immediately sug
gests reduction of wires and cables to the minimum by means of
refinements in design and arrangement . - .
TM 1-320
8 AIR CORPS
.
(10) Struts of strecmWirw form.-It is fou~d in practice that the
best fineness ratio for struts is 4 to 1. Inclining the strut to the vertical
does not have the effect of reducing the resistance for streamline forms,
but for blunter shapes (shorter than the true streamline) inclination
reduces the resistance considerably. A group of strut sections are
shown in figure 14 and the value of K for each shape is shown. It can
be seen that the effect of yawing is to increase greatly the resistanoo by
placing the strut sidewise or at a different angle to the air stream.
(11) AirshVp cars.-All cars are built to take advantage of stream
line form. This is especially true of the inclosed models for which an
average value of K is 0.001. However, there is a wide variance in the
shape of airship cars and a corresponding variance in the value of K.
K=.OOl (average value)
FIGURE 15.-Air stream fiowing by airship~
For each different shape a new value of K must be determined by wind
tunnel test.
c. The following problem illustrates use of the resistance formula:
( 1) Problem.- (a) What is resistance of a fiat . plate 1 foot square
placed at right angles to direction of motion when moving at a velocity
of 30 miles per hour in air of standard density?
(b) What is resistance at 60 miles per hour~
(2) Sol!ution.
(a) Rv= KAV2 =0.00328X 1 X900=2.95 pounds.
(b) Rv = KA V 2 = 0.00328 X 1 X 3600= 11.81 pounds.
This problem illustrates rapidity with which resistance increases
with increasing velocity .
d. Based on resistance of a fiat disk, the following shapes have the
relative resistance shown below :
Percent
Square plate- ----- ------ ----- - --- - - ----- - - ---- ----------- - 104.5
Cylinder, horizontal-- - ------- - ---- ----- --------------- - - 65. 5
Sphere-- -------------------------- ------ ---------------- - 25.4
Cylinder, capped ends------------ - --- - - - - ----------------- 21.0
Airship model -------- - - - ·- -·- --- -------------- ----- 3. 0
AIRSHIP AERODYNAMICS
TM 1-320
9. Coeftioient of skin friction.-a. In the case of a flat plate at
right angles to the air stream the resistance is almost entirely due to
the pressure difference in front of and behind the _plate . This is not
however the case with most solids. In general, resistance may be
divided into two parts:
( 1) Pressure difference.
( 2) Skin friction.
b. When a solid passes through the air it carries along with it a very
thin layer of air, the exterior surface of which forms a plane of air
cleavage. The resistance of the air particles to shear on this plane is
called skin friction.
c. The value of the skin friction on an airship hull, as determined
empirically by Zahm and others, is given by the formula:
R ,= 0.0035pS0 •98 vue
where S is the total surface area. A somewhat more convenient for
mula is--
R r= 0.00309pS vu5
10. Resistance of streamlined body.-a. As mentioned before,
the total resistance is composed of resistanc:A-e-
(1) Caused by pressure difference.
(2) Due to skin friction.
The pressure -difference resistance is least for a very long and slender
form. In fact, the greater the fineness ratio, the less will be the pres
sure-difference resistance. An increase in fineness ratio, however, leads
to an increase in surface area and so to an increase in skin friction . It
is necessary therefore to compromise on a moderate fineness ratio, as
a very long and slender form would have so high a skin friction as to
more than counterbalance the gain by reduction of the pressure
difference resistance. A fl_neness ratio of 4 to 1 is very good for a small
nonrigid, but for large rigid s it has been found advisable to increase
this ratio to 6 or 7 to 1. Recently an airship had been designed whose
hull has a much smaller fineness ratio than the conventional designs.
This airship has a capacity of 200,000 cubic feet and a fineness ratio of
2.82, noticeably shorter than any ships recently constructed. A model
of this ship was tested in the wind tunnel of the Washington Navy
Yard and was found to have the lowest resistance coefficient -of any
model ever tested there.
b. Since the volume varies as the cube of a linear dimension, while
the cross-sectional area and surface area both vary only as the square,
TM 1-3'20
10-11 AIR CORPS
the resistance is proportional to the two-thirds power of the volume.
TMs leads to a more convenient expression for the resistance of airship
hull s as follows :
R = 0 DP (volume) •;s v~.se
where OD is called the P randtl shape coefficient afte r the eminent au
thority, Professor Prandt l. Values of 0 D for various speeds are given
in table I.
c. The offsets for different types of airships are given in table II.
A study of the shapes given therein in connection with the Prandtl
coefficients will bring out the relative efficiency of the different stream
lines.
d. Certain general rules of design developed by experience and test
may be summarized as follows:
( 1) The best form is one of continuous curvature with radius of cur
vature constantly increasing toward rear portion.
(2) T he shape of extreme rear portion of the hull does not seriously
affect the resistance.
( 3) The introduction of a cylin drical midsection causes an addi
tional resistance equal to the skin friction on the increased surface
area of the hull.
(4) The major diameter should lie between 33 and 40 percent of total
length from the bow.
11. Prismatic coefficient. - The ratio of the volume of any hull
form to that of the circumscribing cylinde r is called the prismatic
coefficient, Qv.
Volume
Qv= Maximum cross-sectional area X length
VOl= Q.t.A.L
The prismatic coefficients for different shapes are given in table I . .
~
---. --·- -1 --
"' -t
....
C)
Name of model I x.en ~ th. Dlame·
, I ter. /)
I .... ....
- ·- -. -- I
}
N a.vy B (Goodrich)--- ·_. 3. 5
Navy 0 --- ------- ---- -2. 9
Navy E-------- -- -- -- - 4. 1 E. P- _ .. __ ____ -- - __ - _- 3. 0
I. E _______ -- -- - - - - ---- 2. 9 Goodyear 4 2 _________ 3. 1
Goodyear - L ________ __ 3. 4
Goodyear - 2 ___ ___ - - --- - 3. 8
Goodyear - 3 ____ .. ___ _ . _ 3. 6
Goodyear - 4 ____ -·- ··-- __ 3. 1
~ Astra- Torres __ . . _____ .. _ 3. 1
Ol Parseval P. L __________ 3. 9
Parseval P. !!_ ____ ___ _ _ 3. 2
Parseval P. IlL _______ _ 3. 2
Parseva l S. S. T ___ ___ __ 5. 6
P ony Blimp AA ________ 1. 9
UB-FC ______________ 4. 9
UB- 2 ______________ __ 4. 4
C class cylindric midships
1-1 diameter __ ________ __ 3. 1,
Yz diameter ____ ----- - __ 3. 2 1 diameter _______ ______ 3. 5
2 cliamet3r _____________ 4. 2
3 diamete r ____________ _ 4. 8
4 diameter_ ____________ 5. 5
Feet
0. 6967
. 6417
• 6417
. 6417
. 6417
• 6870
• 6660
• 6350
• 6150
. 6870
. 6914
. 6417
• 6417
• 6417
1. 1330
58"" • ~) V
~ 1. 0591
ll. 1638
- 6417
. 6417
. 6 417
. 6417
. 6 417
. 6417
TABLE I. - Airship model characteristics and data
Area Prandtl sha8~ coefficient, Fine- Dis- Pris-
maxi· ness tance Dis- matic
mum ratio, maxi- tance coeffi· Surface. Volum e,
s cross- Vol. FR mum CG cient,
sectional L diam eter from Q=- VtJI.
area 20 40 60 15 from nose
A m.p.h. m. p.h . m.p.h. nose A XL
-· -- -- - --·····- - -
Sq.ft. Sq.ft. Ou.ft. P.d. L P.ct . L
5. 800 0. 381 0. 8304 0. 0168 o. 0 154 0.0148 5. 060 37. 80 ------ 0. 6176
4. 750 . 323 . 6259 • 0159 . 0144 . 0136 4. 620 30. 00 46.37 . 656 2
5. 007 . 323 6690 . 0168 . 0146 . 0142 4. 870 36. 25 48. 64 . 6621
4. 597 . 323 . 5890 . 0166 . 0147 . 0138 4. 820 41. 59 43. 92 . 6891
4. 597 . 323 • 5955 . 0175 . 0155 . 0144 4. 650 38. 18 44. 25 . 6169
5. 470 . 371 . 7840 • 0162 . 0144 . 0134 4. 640 28. 76 - . .. -- - . 6624
5. 600 . 348 . 7360 ------ --- --- . 0141 5. 130 34. 15 -- ·· - - - . 6184
6. 000 . 317 . 7520 ------ ----- - . 0141 6. 030 36. 14 ---- -- . 6194
5. 900 . 297 . 7760 ------ ------ . 0140 5. 970 36. 36 --~·--- . 7119
5. 470 . 371 . 7840 ------ ------ . 01 53 4. 640 28. 76 ------ . 6624
5. 190 . 309 . 6583 . 0190 . 0159 . 0147 4. 580 33. 80 49. 08 . 6590
5. 465 . 323 . 7240 . 0185 . 0174 . 0165 6. 140 38. 75 43. 19 . 5679 .
4. 528 . 323 . 5891 . 0181 . 0170 . 0164 4. 990 38. 90 44. 46 . 5677
4. 750 . S23 . 6331 . 0179 . 0169 . 0161 4. 699 47. 33 45. 85 . 6095
14.720 1. 008 3. 4550 . 0174 . 0173 . 0170 4. 960 45. 00 45. 88 . 6090
2. 760 . 267 . 3196 . 0205 . 0254 . 0277 3. 410 42. 50 46. 00 . 6003
12. 958 4 . 8810 2. 8603 . 0321 . 0223 . 0219 4. 663 - -· - ~ -- -- ·- ·-... . 65U 6
12. 224C 1. 063::: 2. 9201 . 0205 • 0 189 . 0192 3. 823 - - ....... ----.. ·-- . 61145
.
5. 073 . Z~3 . 6777 . 0154 . 0140 . 013Z 4. 85::: -··-·· .. ... ··-·- - -- 6749
5. 398 . 323 . 7297 . 0153 . 0141 . 0135 5. 100 • . 6909 -- ---- --·- - --
6. 043 . 323 . 8330 . 0164 . 0146 . 0136 5. 570 ------ ------ . 7184
7. 337 . 323 1. 0404 . 0175 . 0150 . 0136 6. 600 ---- -- ------ . 76 11
8. 627 . 323 1. 2471 . 0173 . 0156 . 0148 7. 590 ------ ------ . 7925
9. 922 . 323 1. 4548 . 0175 . 01.'>7 . 0146 8. 590 ------ ------ . 8167
5 diameter _____________ 6. 1 . 641711.218 • 323 1. 6625 • 0164 • 0154 • 0148 9. 602 ----- - -- ---- • 8358
Index ofform efficiency;
Q Hr=- · CD . ---- - ·- .
20 40 60
m.p.h. m. p. h. m. p. h,
36. 76 40. 10 41. 7~
41 27 45. 57 48. 25
- - - --- ------ ------35. 49 40. 08 42. 70
35. 25 39. 80 42. 84
40. 89 45. 37 49. 43
-- -- -- --- --- ---. ...
---- -- ------ -----·· ------------ ----- · ------ -- --- ·~ --~- - -
- 34. 68 41. 45 44. 83
30. 70 32. 64 34. 42
31. 36 33. 39 34. 62
34. 05 36. 06 37. 86
35. 00 35. 23 35. 82
29. 28 23. 63 21 67
- ~~- - - -··- --- --- ---
- -···-- - --- · -- ·-----""'
43. 82 48. 21 51. 13
45. 16 49. 00 51. 18
43. 80 49. 21 52. 82
43. 49 50. 74 55. 96
45. 81 50. 80 53. 55
46. 67 52. 02 55. 94
50. 96 54. 27 56. 47
'!
l:rl
~
~
~ >
~ >-<1
~
~-'t-4
..... ~
t.:l
TABLE I .-Airship model characteristics and data-Continued
.
Area Prandtl shape coefficient Fin&- Dis- . Pris-maxi- ness tanoo Dis- matic
mum CD ratio, maxi- tan co coom-Lenzth· Diame- Surface, Volume, Name of model cross- FR mum CG cient, ter, D 8 sectional Vol. L diameter from Q Vol. area 20 10 60 ~-· from nose D -A-L A m. p. b. m. p. b. m. p. b. nose
- - - -·· ·- - -.
EUiptical series (British)
Feet Feet Sq. ft. Sq. ft. Cu. ft. P.ct.L P.ct. L
E ~ ---------------- ---- 2. 371 0. 3906 ------- 0. 120 0. 1658 0. 0132 0. 0135 -- --- - 6. 070 33. 19 ---- -- 0 5835
E 2 - -- - ------------- -- 1. 743 . 3910 ------- . 120 . 1261 . 0138 . 0128 ------ 4. 460 33. 86 ----- - . 6024
E 3-- -- -------------- - 1. 568 • 3920 ------- . 121 . 1112 . 0147 . 0120 ------ 4. 000 34. 19 ---- - - . 5876 E4 ________________ ___ 1. 384 • 3923 --- ---- . 121 . 0972 . 0167 . 0139 -- ---- 3. 500 35. 18 ---- -- . 5810
~ E 5----- -- ----- -- -- -- - 1. 178 • 3929 ------- . 121 . 0826
'
. 0184 . 0147 -- --- - 3. 000 33. 43 -- ---- . 5786
Parabolic series (British)
p } ___________________ 1. 594 . 3900 ------- . 120 . 0970 . 0168 . 0137 -- --- - 4. 090 49. 39 -- ---- . 5094 P2 ______________ ____ _ 1. 598 . 3903 ------- - 120 . 1000 . 0169 . 0176 -- ---- 4. 070 32. 06 ---- -- . 5265 Pa __________ _________ 1. 173 . 3867 ---- -- - . 117 . 0729 . 0226 . 0173 ------ 3. 830 50. 35 ------ . 5293
P4- ------- --- ------- - 1. 217 . 3870 ------- . 118 • 0714 . 02-15 . 0193 -- --- - 3. 140 35. 05 ------ . 4989
- --···- -------------------------- -··- - ---- ---------------
..
Index of form efficiency
Q H,--CD
20 40 eo m.p.h. m. p.h. m. p. 11.
- -
44. 20 43. 22 ------43. 65 47. 06 --- ---40. 00 45. 55 -- ----34. 79 41. 80 ---- --31. 45 39. 36 ------
30. 32 37. 18 ------31. 15 30. 00 ---- --23. 42 30. 60 ------23. 20 25. 85 -- ----
···---------- ------ --- ----
~
... I-'
... J.,
~
~ al
t-:)