CONSIDERATION OF SIZE AND SPEED OF VEHICLES

Dimensions, weight or mass, cruising speed and density of the medium through which they move, all have an influ­ence upon the geometrical design of dynamically lifted vehicles and their operational qualities.

Square-Cube Law. The volume (weight, mass) of a body grows in proportion to the cube of its linear dimension £ , while its significant area is only S ^((1. As an ex­ample, we may compare the frontal area of a motorcycle (with 1 passenger) with that of a bus (with 50 passengers) or a railroad train (with 500 passengers). It can be found that the aerodynamic drag, roughly proportional to frontal area, is “very” high per passenger for the motor­cycle, and “very” low per passenger for the train. Aero­dynamic efficiency of these land vehicles, thus increases with their size. In the case of an airplane, the significant area is that of the wing (lifting the vehicle), while the volume is the equivalent of its mass, figure 7. Therefore, the mass grows “faster” with size ‘je than the areas of its lifting, stabilizing and controlling surfaces.

Froude Number. In the hydrodynamic design of ships, the Froude number is used to establish similarity between to wing-tank tests and full-scale operation, primarily in regard to wave pattern and wave resistance in calm water (19). The square of this number is the ratio between the inertial or wave-producing forces involved, and the weight (due to gravity) of water and/or vessel. Thus:

Подпись:= гУЛг103 =v*lsi (41)

where p^= )£/g, XZr= weight per unit volume (v=j) of the water displaced by the ship’s hull (displaced volume of water). Note that the vehicle’s weight is W = $ f,

where )T (subscript v) is equal to that of the water (sub­script w).

CONSIDERATION OF SIZE AND SPEED OF VEHICLES

Figure 7. Demonstration of the influence of size and speed upon the relative dimensions of wing and volume of airplanes or hydro­foil boats.

(17) Water tunnels, some of them used for “aerodynamic” investi­gations:

a) See author’s description and results in Fieseler Rpts (1939, 1940).

b) Hoerner, Fieseler Water Tunnel, Yearbk D Lufo 1943 (not distributed).

c) Drescher, Water Tunnels of the AVA, Yearbk D Lufo 1941 p 1-714.

d) Ross-Robertson, 4-Foot Tunnel Penn University, Trans SNAME 1948 p 5.

e) Brownell, Variable Pressure Tunnel, DTMB Rpt 103 2 (1956); see also description of tunnels in TMB Rpt 1856 (AD-607, 773).

(18) Introduction of Froude number in this book, based upon:

a) GabrieUi and von Karman (Mech Engg 1950 p 775, ;:r J. ASNE 1951 p 188).

b) Davidson (Stevens ETT TM 97 & Note 154, 1951; also SNAME Bulletin 1955).

c) Hoerner, Consideration of Size-Speed-Power in Hydrofoil Craft, SAE Paper 5 22-B (1962); reprinted in Naval Engineers Journal 1963 p 915.

d) Froude, Collected Papers and Memoir, Inst Nav Architects (London) 1955. [9]

Hydrofoil Boats. In a boat supported above the water, by “wings” running below the surface, weight or “load” is concentrated upon comparatively small foils. The density of these foils in terms of W/S, or m/S (in kg/m2) is “very” high. Neglecting the comparatively small surface waves left behind these boats when “flying”, the inertial forces are now fluid-dynamic lift (and induced drag). The Froude number may be written in the form of

(v*/gf) or (УЇІЇЛ) or (VK/$7T)

where к is indicating speed in knots, and A = weight of the craft in long tons (representing its mass). These num­bers have a definite influence upon the geometric con­figuration of hull and foils. Since it is desirable to limit the foil span to the dimension of the hull’s beam, the following similarity numbers are statistically found:

(V/v3T~) = for configurations with 3 foils

= for “wing” plus stabilizing foil = for a pair of foils in tandem

While it is possible to make a boat with 3 foils very small, there is an upper limit to the size of operationally feasible hydrofoil boats. The size (span) of the foils required, simply outgrows that of the hull (beam). For a speed below 50 knots, therefore, the largest boat practicable may be in the order of 400 long tons (some 400 metric tons).

Lift Coefficient. When comparing airplanes with each oth­er, interpretation of the Froude number is as follows. The fluid-dynamic force (lift) acting upon the “vehicle” is proportional to (pV S), while the number reflects weight and/or mass forces of the vehicle (subscript ‘V’):

(p/ > = S)/(m/g) = (fj fr) V[10]/g 2 (42)

where the significant area S = і is that of the wing. The subscript ‘a’ indicates ‘ambient’, while is the density of the airplane (in kg/m ). This type of Froude number thus contains a density ratio. The mass density of a conven­tional airplane may be m/V = 200 (kg/m[11] ), in comparison to 1000 for water, and roughly 1.0 for air (at an altitude of 7 km). The density ratio for an airplane is in the order of 200, accordingly. To support the craft by means of aerodynamic lift, the speed in (pV2^2) has to be much higher than that of a hydrofoil boat, say between 200 and 800, instead of 40 knots, for example. Equation (42) can now be rewritten as

(m/g)/( faV2S) = W/^V2S = 2 CL (43)

In other words, the lift coefficient

CL = 2(W/ ?.„V2S) = 2/F/ (44)

is the equivalent of a Froude number. Table 1 has been prepared, containing characteristics of several typical, but extremely different lift-supported vehicles (including birds). It can tenatively be assumed that all of them fly (cruise, climb, keep aloft) at the “same” lift coefficient, say between 0.3 and 0.7.

STOL — airplanes necessarily need to have exceptionally high lift coefficients (possibly up to 8). One example is included in Table 1 on “size and speed”. Structurally, such aircraft must be expected to be limited in size (wing area and span). High lift coefficients are obtained by trailing edge flaps, by propeller-slipstream deflection, and finally by direct application of thrust (as in helicopters, for example). Figure 8 demonstrates how in helicopters, VTOL and STOL aircraft lift is ideally produced, by deflecting a stream tube of air or by direct downward deflection.

TABLE I

Table I, average characteristics (in rough and round num­bers) of various lift-supported “vehicles”, including birds.

m

= mass

(kg)

ft

II

3^

>■0

u

kg/m3

S

= wing area

(m2)

4

= speed

knots

b

= Span

(m)

At

= weight

l’tons

“vehicle”

W(kp)

b(m)

S(m2)

(m)

W/S

VK

Vfit

V(m/s)

cL

РЛ

*7

buzzard (22)

l

1

0.2

0.4

4

18

90

9

0.7

8

3

albatross (22)

8

3

1

1

8

33

130

17

0.5

8

4

small airplane

1,000

10

20

4

50

130

200

65

0.4

11

9

STOL airplane

5,000

20

35

6

140

40

3

21

5.0

20

4

fighter airplane

8,000

18

18

4

440

520

370

260

0.2

160

10

C5-A airplane

250,000

60

600

22

420

480

190

240

0.3

50

10

hydrofoil boat

100,000

10

25

5

4000

40

19

20

0.3

1

8

High Altitude means low density. The value of density at 10 km (30,000 ft, where airliners obtain their longest range) is roughly 40% of that at sea-level. Similarity in the geometric design (wing area in comparison to fuselage dimensions) of two airplanes, one designed for low, arid the other for high altitude (say 20 km as the SST, where density is only some 7% of that at sea level) can only be maintained when keeping the Froude number as per equa­tion (42) constant. A larger airplane will statistically have to be designed for a higher speed; and when flying at higher altitude, the speed has to be higher again. These considerations lead to supersonic cruising speeds (as in the SST).

Control High lift coefficients are usually produced by part-span flaps, while the tail surfaces and particularly the ailerons of an airplane, remain unchanged. Considering a rolling STOL, similar to that in the “size-speed’’ table, but flying at Ct_ = 0.8 for example, the control ‘power’ corre­sponds to the low value F/ = 2.5. Control effectiveness would Thus be only lA that of the fighter airplane. It can also be said that the control surfaces, or their area ratio (control/wing) should be 4 times as high to obtain the same result as with the fighter. The low Froude numbers of the birds as in the table, suggest that they have a highly effective automatic control system, in the form of feel, instant reaction and muscles serving as actuators. They also have variable areas in their wings and tails; and they can twist all surfaces to a degree not found in conven­tional man-made airplanes.

Dynamic Stability. The motions of an airplane, resulting from the combination of fluid-dynamic forces (such as in the tail surfaces, for example) and forces arising in the masses of the craft as a consequence of acceleration (and deceleration) determines the dynamic characteristics (19). The term in the equations of motions accounting for these forces is the relative density, or the ratio of the mass of the vehicle to that of the fluid affected. Therefore, the Froude number as in equation 42 could directly be used when determining dynamic stability. For example, oscill­atory motions of an airplane will be “similar” to those of a properly built wind-tunnel model, when the Froude number is kept constant. On the basis of the F’numbers in Table 1, it can also be expected that a fighter-type air­plane can be very stable.

Advanced Vehicles. Table 1 in this section only presents examples of birds, airplanes and a hydrofoil boat. In reality, there is a wide range of size and speed in each category of vehicle, depending among others, upon power or thrust installed. For example, a fighter airplane is designed for high speed; its Froude number is particularly high, accordingly. Nevertheless, there are statistical trends evident; the average lift coefficient decreases as the size is increased; the Froude number increases as the speed is increased. The speed-size relation is not usually considered when designing an airplane. Power available and speed are

CONSIDERATION OF SIZE AND SPEED OF VEHICLES

(a) Lift produced in wing bу defleotion of a atreaa tube with the effeotlT* diameter equal to the wing span.

CONSIDERATION OF SIZE AND SPEED OF VEHICLES

Figure 8. The origin of lift:

a) produced in a wing by deflecting a streamtube of air,

b) produced by a fan, propeller or helicopter rotor.

a matter of specifications for a particular type of airplane required. To produce this aircraft, is then a matter of structural design, engine development, the introduction of new materials. For ехдтріе, the increase of maximum lift from CL)< = 1.5 (say around 1930) to 7.5 as in STOL airplanes (around 1960 or 1965) is made possible by elaborate structural innovations, such as triple-slotted wing flaps and tilting wing-engine-propeller combinations.

(20) Calculation of airplane performance:

a) Oswald, Performance Formulas and Charts, NACA Rpt 408 (1932).

b) Diehl, in “Engineering Aerodynamics”, Ronald New York, 1928 to 1936.

c) Wood, “Technical Aerodynamics”, McGraw-Hill (1935, 1947) by author (1955).

d) Perkins-Hage, “Airplane Performance Stability Control”, Wiley 1949.

e) Dommasch, “Airplane Aerodynamics”, Pitman 1951.

0 Wood, “Aircraft Design”, Johnson Publishing (Boulder, Colo.) 1963.

g) Breguet, Endurance (1921); see J. Aeron Sci 1938 p 436; see (b).

(21) Characteristics and Performance of KC-135 (Boeing 707):

a) Vancey, КС-135 Flight Tests, Edwards AF Base TR-1958-26; AD-152, 257.

b) Tambor, Flight Tested Lift and Drag, NASA TN D-30 (1960).

Подпись:It is not yet clear where the consideration of size and speed similarity (or dissimilarity) can be of practical value. One example in this respect seems to be the control of aircraft. In very high altitudes (low air density, as in NASA’s more or less ballistic X-15) and/or at low speeds (as in STOL airplanes) aerodynamic devices are no longer sufficient. Suitably located gas jets, coupled with auto­matic sensing and actuating devices, therefore, have to be used. An awareness of the influence of size, speed, densi­ty; and, of course, the anticipation of limitations or diffi­culties such as supersonic effects (or cavitation in water) may prepare the designer for the effort in research and development required, before proceeding with the con­struction of any advanced vehicle, or the introduction of a new mode of lift-supported transportation.

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