Chapter 1 — Part 2

DEVELOPMENT OF BOUNDARY L~AYER

ON A SMOOTH FLAT PLATE

TURBULENT

BOUNDARY

LLAMINAR

SUB-LAYER

COMPARISON OF VELOCITY PROFILES

FOR LAMINAR AND TURBULENT BOUNDARY LAYERS

TURBULENT

PROFILE

LAMINAR

PROFILE

- LOW THICKNESS - GREATER THICKNESS

- LOW VELOCITIES NEXT TO SURFACE - HIGHER VELOCITIES NEXT TO SURFACE

- GRADUAL VELOCITY CHANGE - SHARP VELOCITY CHANGE

- LOW SKIN FRICTION - HIGHER SKIN FRICTION

figure 7.24. Boundary Layer Charactorisfics

NAVWEPS CO-SOT-80

BASIC AE,RODYNAMlCS

the turbulent area, a sharp “crackling” will

be audible.

In order to compare the characteristics of

the laminar and turbulent boundary layers, the

velocity profiles (the variation of boundary

layer velocity with height above the surface)

should be compared under conditions which

could produce either laminar or turbulent

flow. The typical laminar and turbulent pro-

files are shown in figure 1.24. The velocity

profile of the turbulent boundary layer shows

a much sharper initial change of velocity but

a greater height (or boundary layer thickness)

required to reach the free stream velocity.

As a result of these differences, a comparison

will show:

(1) The turbulent boundary layer has a

fuller velocity profile and has higher local

velocities immediately adjacent to the sur-

face. The turbulent boundary layer has

higher kinetic energy in the airflow next to

the surface.

(2) At the surface, the laminar boundary

layer has the less rapid change of velocity

with distance above the plate. Since the

shearing stress is proportional to the velocity

gradient, the lower velocity gradient of the

laminar boundary layer is evidence of a

lower friction drag on the surface. If the

conditions of flow were such that either a

turbulent or a laminar boundary layer could

exist, the laminar skin friction would be

about one-third that for turbulent flow.

The low friction drag of the laminar bound-

ary layer makes it quite desirable. However,

the transition tends to take place in a natural

fashion and limit the extensive development

of the laminar boundary layer.

REYNOLDS NUMBER. Whether a lam-

inar or turbulent boundary layer exists depends

on the combined effects of velocity, viscosity,

distance from the leading edge, density, etc.

The effect of the most important factors is

combined in a dimensionless parameter called

“Reynolds Number, RN.” The Reynolds

Number is a dimensionless ratio which por-

trays the relative magnitude of dynamic and

viscous forces in the flow.

where

RiV=Reynolds Number, dimensionless

V= velocity, ft. per sec.

x= distance from leading edge, ft.

Y= kinematic viscosity, sq. ft. per sec.

While the actual magnitude of the Reynolds

Number has no physical significance, the

quantity is used as an index to predict and

correlate various phenomena of viscous fluid,

flow. When the RN is low, viscous or fric-

tion forces predominate; when the RN is high,

dynamic or inertia forces predominate. The

effect of the variables in the equation for

Reynolds Number should be understood. The

RN varies directly with velocity and distance

back from the leading edge and inversely with

kinematic viscosity. High RN’s are obtained

with large chord surfaces, high velocities, and

low altitude; low RN’sresult from small chord

surfaces, low velocities, and high altitudes-

high altitudes producing high values for kine-

matic viscosity.

The most direct use of Reynolds Number is

the indexing or correlating the skin friction

drag of a surface. Figure 1.25 illustrates the

variation of the friction drag of a smooth,

flat plate with a Reynolds Number which is

based on the length or chord of the plate.

The graph shows separate lines of drag coeffi-

cient if the flow should be entirely laminar or

entirely turbulent. The two curves for lam-

inar and turbulent friction drag illustrate the

relative magnitude of friction drag coefficient

if either type of boundary layer could exist.

The drag coefficients for either laminar or tur-

bulent flow decrease with increasing RN since

the velocity gradient decreases as the boundary

layer thickens.

NAWWEPS OD-EOT-SO

BASIC AERODYfflAMICS

FRICTION DRAG OF A SMOOTH

FLAT PLATE

c ,020 -

E D ,010 -

iii .008 -

yu’ .%2 -

O” 0 :% -

:: ,002 - ‘\

2i ‘1

.OOl * 1 I 1 1 1

0.1 0.5 1.0 5.0 10.0 50 100

REYNOLDS NUMBER

RN(MILLIONS)

CONVENTIONAL AfdD LAMINAR

FLOW SECTIONS

TRANSITION

NACA /

L NACA 0009

P DRAG BUCKET”

I I

-1.0 -3 0 .5 I.0 I.§

SECTION LIFT COEFFICIENT, cl

Figure 7.25. Skin Friction Drag

Weaised January 1965

NAVWEPS 00-SOT-80

BASIC AERODYNAMICS

If the surface of the plate is smooth and the

original airstream has no turbulence, the plate

at low Reynolds Numbers will exist with pure

laminar flow. When the RN is increased to

approximately 530,000, transition occurs on

the plate and the flow is partly turbulent.

Once transition takes place, the drag coefficient

of the plate increases from the laminar curve

to the turbulent curve. As the RN approaches

very high values (20 to 50 million) the drag

curve of the plate approaches and nearly equals

the values for the turbulent curve. At such

high RN the boundary layer is predominantly

turbulent with very little laminar flow-the

transition point is very close to the leading

edge. While the smooth, flat plate is not ex-

actly representative of the typical airfoil, basic

fluid friction phenomena are illustrated. At

RN less than a half million the boundary layer

will be entirely laminar unless there is extreme

surface roughness or turbulence induced in the

airstream. Reynolds Numbers between one

and five million produce boundary layer flow

which is partly laminar and partly turbulent.

At RN above ten million the boundary layer

characteristics are predominantly turbulent.

In order to obtain low drag sections, the

transition from laminar to turbulent must be

delayed so that a greater portion of the sur-

face will be influenced by the laminar bound-

ary layer. The conventional, low speed air-

foil shapes are characterized, by minimum

pressure points very close to the leading edge.

Since high local velocities enhance early

transition, very little surface is covered by

the laminar boundary layer, A comparison

of two 9 percent thick symmetrical airfoils is

presented in figure 1.25. One section is the

“conventional” NACA C!UO~ section which

has a minimum pressure point at approxi-

mately 10 percent chord at zero lift. The other

section is the NACA 66039 which has a

minimum pressure point at approximately 60

percent chord at zero lift. The lower local

velocities at the leading edge and the favor-

able pressure gradient of the NACA 66-009

delay the transition to some point farther aft

on the chord. The subsequent reduction in

friction drag at the low angles of attack ac-

counts for the “drag bucket” shown on the

graphs of cd and cI for these sections. Of

course, the advantages of the laminar flow

airfoil are apparent only for the smooth airfoil

since surface roughness or waviness may pre-

clude extensive development of a laminar

boundary layer.

AIRFLOW SEPARATION. The character

of the boundary layer on an aerodynamic

surface is greatly influenced by the pressure

gradient. In order to study this effect, the

pressure distribution of a cylinder in a perfect

fluid is repeated in figure 1.26. The airflows

depict a local velocity of !zero at the forward

stagnation point and a maximum local velocity

at the extreme surface. The airflow moves

from the high positive pressure to the minimum

pressure point-a favorable pressure gradient

(high to low). As the air moves from the

extreme surface aft, the local velocity decreases

to zero at the aft stagnation point. The static

pressure increases from the minimum (or max-

imum suction) to the high positive pressure

at the aft stagnation point-an adverse pres-

sure gradient (low to high).

The action of the pressure gradient is such

that the favorable pressure gradient assists

the boundary layer while the adverse pressure

gradient impedes the flow of the boundary

layer. The effect of an adverse pressure gradi-

ent is illustrated by the segment X-Y of figure

1.26. A corollary of the skin friction drag is

the continual reduction of boundary layer

energy as flow continues aft on a surface. * The

velocity profiles of the boundary layer are

shown on segment X-Y of figure 1.26. In the

area of adverse pressure gradient the bound-

ary layer flow is impeded and tends to show a

reduction in velocity next to the surface. If

the boundary layer does not have sufhcient

kinetic energy in the presence of the adverse

pressure gradient, the lower levels of the

boundary layer may stagnate prematurely.

NO SEPARATION

NAWWEPS 00-8OT-80

BASIC AERODYNAMICS

SEPARATION 1

BOUNDARY LAYER

SEPAF --‘-.’ iAT ION /-------

SEPARATION AT STALL

REVERSE

FLOW

b SHOCK WAVE

SHOCK WAVE INDUCED

FLOW SEPARATION

Figure 1.26. Airflow Separation (sheet 7 of 2)

Figure 7.26. Airflow Separation (sheet 2 of 2)

Premature stagnation of the boundary layer

means that all subsequent airflow will overrun

this point and the boundary layer will separate

from the surface. Surface flow which is aft of

the separation point will indicate a local flow

direction forward toward theseparation point-

a flow reversal. If separation occurs the posi-

tive pressures are not recovered and form drag

results. The points of separation on any aero-

dynamic surface may be noted by the reverse

flow area. Tufts of cloth or string tacked to

the surface will lie streamlined in an area of

unseparated flow but will lie forward in an

area behind the separation point.

The basic feature of airflow separation is

stagnation of the lower levels of the boundary

layer. Airjh ~cparation muh when the lower

lcvcls of the boundary layer do not have sujicicnt

kinetic cncrgy in the prwncc of an advcm ps.wrc

gradient. The most outstanding cases of air-

flow separation are shown in figure 1.26. An

airfoil at some high angle of attack creates a

pressure gradient on the upper surface too

severe to allow the boundary layer to adhere

to the surface. When the airflow does not

adhere to the surface near the leading edge

the high suction pressures are lost and stall

occurs. When the shock wave forms on the

upper surface of a wing at high subsonic speeds,

the increase of static pressure through the

shock’ wave creates a very strong obstacle for

the boundary layer. If the shock wave is

sufhciently strong, separation will follow and

“compressibility buffet” will result from the

turbulent wake or separated flow.

In order to prevent separation of a boundary

layer in the presence of an adverse pressure

gradient, the boundary layer must have the

highest possible kinetic energy. If a choice is

available, the turbulent boundary layer would

be preferable to the laminar boundary layer

because the turbulent velocity profile shows

higher local velocities next to the surface.

The most effective high lift devices (slots,

slotted flaps, BLC) utilize various techniques

NAVWEPS OO-SOT-80

BASIC AERODYNAMICS

to increase the kinetic energy of the upper sur-

face boundary layer to withstand the more

severe pressure gradients common to the higher

lift coefficients. Extreme surface roughness

on full scale aircraft (due to surface damage,

heavy frost, etc.) causes higher skin friction

and greater energy loss in the boundary layer.

The lower energy boundary layer may cause a

noticeable change in C,-” and stall speed. In

the same sense, vortex generators applied to

the surfaces of a high speed airplane may allay

compressibility buffet to some degree. The

function of the vortex generators is to create a

strong vortex which introduces high velocity,

high energy air next to the surface to reduce

or delay the shock induced separation. These

examples serve as a reminder that separation is

the result of premature stagnation of the

boundary layer-insufficient kinetic energy in

the presence of an adverse pressure gradient.

SCALE EFFECT. Since the boundary layer

friction and kinetic energy are dependent on

the characteristics of the boundary layer,

Reynolds Number is important in correlating

aerodynamic characteristics. The variation of

the aerodynamic characteristics with Reynolds

Number is termed “scale effect” and is ex-

tremely important in correlating wind tunnel

test data of scale models with the actual flight

characteristics of the full size aircraft. The

two most important section characteristics

affected by scale effects are drag and maximum

lift-the effect on pitching moments usually

being negligible. From the known variation

of boundary layer characteristics with Rey-

nolds Number, certain general effects may be

anticipated. With increasing Reynolds Num-

ber, it may be expected that the section maxi-

mum lift coefficient will increase (from the

higher energy turbulent boundary layer) and

that the section drag coefficient will decrease

(similar to that of the smooth plate). These

effects are illustrated by the graphs of figure

1.27.

The characteristics depicted in figure 1.27

are for the NACA 4412 airfoil (4 percent

RN

MILLION

-6.0 11

4 8 12 16 20

SECTION ANGLE OF ATTACK

=o 1 DEGREES

-I-

RN - 1.5 MILLION

I I I 1-

-.5 0 .5 I.0 1.5

SECTION LIFT COEFFICIENT

c.l

figure 1.27. Effect of Reymafds Number on Section Ckacteristics of NACA 4412

camber at 40 percent chord, 12 percent thick-

ness at 30 percent chord)--a fairly typical

“conventionaal” airfoil section. The lift curve

show a steady increase in cl with increasing

RN. However, note that a>maller change in

cr occurs between Reynolds Numbers of 6.0

ad 9.0 million than occurs between 0.1 and

3.0 million. In other words, greater changes

in CI occur in the range of Reynolds Num-

bers zhere the laminar (low energy) boundary

layer predominates. The drag curves for the

section show essentially the same feature-the

greatest variations occur at very low Reynolds

Numbers. Typical full scale Reynolds Num-

bers for aircraft in flight may be 3 to 5@O million

where the boundary layer is predominately

turbulent. Scale model tests may involve

Reynolds Numbers of 0.1 to 5 million where

the boundary layer be predominately laminar.

Hence, the “scale” corrections are very neces-

sary to correlate the principal aerodynamic

characteristics.

The very large changes in aerodynamic

characteristics at low Reynolds Numbers are

due in great part to the low energy laminar

boundary layer typical of low Reynolds Num-

bers. Low Reynolds Numbers are the result

of some combination of low velocity, small

size, and high kinematic viscosity RN= ( 3

Thus, small surfaces, low flight speeds, or very

high altitudes can provide the regime of low

Reynolds Numbers. One interesting phenom-

enon associated with low BN is the high form

drag due to separation of the low energy

boundary layer. The ordinary golf ball oper-

ates at low RN and would have very high

form drag without dimpling. The surface

roughness from dimpling disturbs the laminar

boundary layer forcing a premature transition

to turbulent. The forced turbulence in the

boundary layer reduces the form drag by pro-

viding a higher energy boundary layer to

allay separation. Essentially the same effect

can be produced on a model airplane wing by

roughening the leading edge-the turbulent

NAVWEPS DD-RDT-80

BASIC AERODYNAMICS

boundary layer obtained may reduce the form

drag due to separation. In each instance, the

forced transition will be beneficial if the reduc-

tion in form drag is greater than the increase

in skin friction. Of course, this possibility

exists only at low Reynolds Numbers.

1,n a similar sense, “trip” wires or small

surface protuberances on a wind tunnel model

may be used to force transition of the boundary

layer and simulate the effect of higher Reynolds

Numbers.

PLANFORM EFFECTS AND

AIRPLANE DRAG

EFFECT OF WING PLANFORM

The previous discussion of aerodynamic

forces concerned the properties of airfoil sec-

tions in two-dimensional flow with no consid-

eration given to the influence of the planform.

When the effects of wing planform are intro-

duced, attention must be directed to the ex-

istence of flow components in the spanwise

direction. In other words, airfoil section

properties deal with flow in two dimensions

I while plonform properties consider flow in

three dimensions.

In order to fully describe the planform of a

wing, several terms are required. The terms

having the greatest influence on the aerody-

namic characteristics are illustrated in figure

1.28.

(1) The wing r?rc11, S, is simply the plan

surface area of the wing. Although a por-

tion of the area may be covered by fuselage

or nacelles, the pressure carryover on these

surfaces allows legitimate consideration of

the entire plan area.

(2) The wing ~ptia, 6, is measured tip to

tip.

(3) The avcragc chord, c, is the geometric

average. The product of the span and the

average chord is the wing area (6X6=$).

(4) The aspect ratio, AR, is the proportion

of the span and the average chord.

AR=b/c

NAVWEPS 00-SOT-80

BASIC AERODYNAMICS

p-----y

S= WING AREA, SO. FT.

b= SPAN, FT

c = AVERAGE CHORD, FT

AR = ASPECT RATIO

AR = b/c

AR= b:s

I b ----_I

CR = ROOT CHORO, FT

Ct = TIP CHORD, FT

x = TAPER RATIO

A= SWEEP ANGLE, DEGREES

MAC : MEAN AERODYNAMIC CHORD, FT.

Figure 1.28. Description of Wing Planform

If the planform has curvature and the aver-

age chord is not easily determined, an

alternate expression is:

AR = b2/.S

The aspect ratio is a fineness ratio of the

wing and this quantity is very powerful in

determing the aerodynamic characteristics

and structural weight. Typical aspect ratios

vary from 33 for a high performance sail-

plane to 3.5 for a jet fighter to 1.28 for a

flying saucer.

(5) The raat chord, c,, is the chord at the

wing centerline and the rip chord, c,, is

measured at the tip.

(6) Considering the wing planform to

have straight lines for the leading and trail-

ing edges, the taper ratio, A (lambda), is the

ratio of the tip chord to the root chord.

A=&

The taper ratio affects the lift distribution

and the structural weight of the wing. A

rectangular wing has a taper ratio of 1.0

while the pointed tip delta wing has a taper

ratio of 0.0.

(7) The sweep angle, A (cap lambda), is

usually measured as the angle between the

line of 25 percent chords and a perpendicular

to the root chord. The sweep of a wing

causes definite changes in compressibility,

maximum lift, and stall characteristics.

(8) The mean aerodynamic chord, MAC,

is the chord drawn through the centroid

(geographical center) of plan area. A rec-

tangular wing of this chord and the same

span would have identical pitching moment

characteristics. The MAC is located on the

reference axis of the airplane and is a primary

reference for longitudinal stability considera-

tions. Note that the MAC is not the average

chord but is the chord through the centroid

of area. As an example, the pointed-tip

delta wing with a taper ratio of zero would

have an average chord equal to one-half the

NAVWEPS OO-BOT-BO

BASIC AERODYNAMICS

root chord but an MAC equal to two-thirds

~‘of the root chord.

The aspect ratio, taper ratio, and sweepback

of a planform are the principal factors which

determine the aerodynamic characteristics of a

.wing. These same quantities also have a defi-

nite influence on the structural weight and stiff-

ness of a wing.

DEVELOPMENT OF LIFT BY A WING.

In order to appreciate the effect of the planform

on the aerodynamic characteristics, it is neces-

sary to study the manner in which a wing

produces lift.’ Figure 1.29 illustrates the three-

dimensional flow pattern which results when

the rectangular wing creates lift.

J.f a wing is producing lift, a pressure differ-

ential will exist between the upper and lower

surfaces, i.e., for positive lift, the static pres-

sure on the upper surface will be less than on

the lower surface. At the tips of the wing,

the existence of this pressure differential creates

the spanwise flow components shown in figure

1.29: For the rectangular wing, the lateral

flow developed at the tip is quite strong and a

strong vortex is created at the tip. The lateral

‘flow-and consequent vortex strength-reduces

inboard from the tip until it is zero at the

centerline.

The existence of the tip vortex is described

by the drawings of figure 1.29. The rotational

pressure flow combines with the local airstream

flow to produce the resultant flow of the

trailing vortex. Also, the downwash flow

field behind a delta wing is illustrated by the

photographs of figure 1.29. A tuft-grid is

mounted aft of the wing to visualize the local

flow direction by deflection of th,e tuft ele-

ments. This tuft-grid illustrates the existence

of the tip vortices and the deflected airstream

aft of the wing. Note that an increase in

angle of attack increases lift and increases the

flow deflection and strength of the tip vortices.

Figure 1.30 illustrates the principal effect

of the wing vortex system. The wing pro-

ducing lift can be represented by a series of

NAWWEPS 00-8OT-80

BASIC AERODYNAMICS

WING UPPER SURFACE

TIP VORTEX

WING LOWER SURFACE VORTICES ALONG

TRAILING EDGE

TRAILING EDGE

I/

I UPPER SURFACE

LEADING EDGE FLOW

FLOW

LOW PRESSURE-

,-

HIGH PRESSURE)

Figure 1.29. Wing Three Dimensional Flow (sheet 1 of 2)

Revised January 1965

NAVWEPS OO-BOT-RD

BASIC AERODYNAMICS

DOWNWASH FLOW FIELD BEHIND

A DELTA WING ILLUSTRATED

BY TUFT-GRID PHOTOGRAPHS AT

VARIOUS ANGLES OF ATTACK

--A-- 30” OF FLOW ANGULARITY

“T

(DEG)

I) TUFT GRID 6 INCHES FROM (b) TUFT GRID 24 INCHES FROM

TRAILING EDGE. TRAILING EDGE.

FROM NACA TN 2674

Figure 1.19. Wing Three Dimensional Flow (sheet 2 of 2)

NAVWEPS 00-8OT-80

BASIC AERODYNAMICS

vortex filaments which consist of the tip or

trailing vortices coupled with the bound or

line vortex. The tip vortices are coupled with

the bound vortex when circulation is induced

with lift. The effect of this vortex system is

to create certain vertical velocity components

in the vicinity of the wing. The illustration

of these vertical velocities shows that ahead

of the wing the bound vortex induces an up-

wash. Behind the wing, the coupled action

of the bound vortex and the tip vortices in-

duces a downwash. With the action of tip

and bound vortices coupled, a final vertical

velocity (220) is imparted to the airstream by

the wing producing lift. This result is an

inevitable consequence of a finite wing pro-

ducing lift. The wing Producing lift applies

the equal and opposite force to the airstream

and deflects it downward. One of the impor-

tant factors in this system is that a downward

velocity is created at the aerodynamic center

(w) which is one half the final downward

velocity imparted to the airstream (2~).

The effect of the vertical velocities in the

vicinity of the wing is best appreciated when

they are added vectorially to the airstream

velocity. The remote free stream well ahead

of the wing is unaffected and its direction is

opposite the flight path of the airplane. ‘Aft

of the wing, the vertical velocity (2~) adds to

the airstream velocity to produce the down-

wash angle e (epsilon). At the aerodynamic

center of the wing, the vertical,velocity (w)

adds to the airstream velocity to produce a

downward deflection of the airstream one-half

that of the downwash angle. In other words,

the wing producing lift by the deflection of an

airstream incurs a downward slant co the wind

in the immediate vicinity of the wing. Hence,

the JeCtionJ of the wing operate in an average rela-

tive wind which is inclined downward one-half the

final dowraw& angle. This is one important

feature which distinguishes the aerodynamic

properties of a wing from the aerodynamic

properties of an airfoil section.

The induced velocities existing at the aero-

dynamic center of a finite wing create an aver-

age relative wind which is different from the

remote free stream wind. Since the aerody-

namic forces created by the airfoil sections of a

wing depend upon the immediate airstream in

which they operate, consideration must be

given to the effect of the inclined average rela-

tive wind.

To create a certain lift coefficient with the

airfoil section, a certain angle must exist be-

tween the airfoil chord line and the avcragc

relative wind. This angle of attack is a,,, the

section angle of attack. However, as this lift

is developed on the wing, downwash is in-

curred and the average relative wind is in-

clined. Thus, the wing must be given some

angle attack greater than the required section

angle of attack to account for the inclination of

the average relative wind. Since the wing

must be given this additional angle of attack

because of the induced flow, the angle between

the average reiative wind arid tlie remote fiCC

stream is termed the induced angle of attack,

ai. From this influence, the wing angle of

attack is the sum of the section and induced

angles of attack.

a=ul)+a;

where a= wing angle of attack

OLD= section angle of attack

OI;= induced angle of attack

INDUCED DRAG

Another important influence of the induced

flow is the orientation of the actual lift on a

wing. Figure 1.30 illustrates the fact that the

lift produced by the wing sections is perpen-

dicular to the average relative wind. Since

the average relative wind is inclined down-

ward, the section lift is inclined aft, by the

same amount-the induced angle of attack,

ai. The lift and drag of a wing must continue

to be referred perpendicular and parallel to the

remote free stream ahead of the wing. In this

respect, the lift on the wing has a component

of force parallel to the remote free stream.

This component of lift in the drag direction

is the undesirable-but unavoidable-conse-

NAVWEPS DD-ROT-80

BASIC AERODYNAMICS

BOUND OR :INE VORTEX

, OR TIP VORTEX

DEFLECTED AIRSTREAM

(UPW

BOUND VORTEX ONLY

VERTICAL VELOCITIES

IN THE VICINITY OF

THE WING

COUPLED BOUND AND

AVERAGE RELATIVE WIND TIP VORTICES

V t

REMOTE FREE STREAM

AT WING A.C.

DOWNWASH

ANGLE

D it INDUCED DRAG

EFFECTIVE

LIFT-

REMOTE FREE STREAM

Figure 1.30. Wing Vortex System and Induced Flow

NAVWEPS OO-SOT-~O

BASIC AERODYNAMICS

quence of developing lift with a finite wing

and is termed INDUCED DRAG, D+ In-

duced drag is separate from the drag due to

form and friction and is due simply to the de-

velopment of lift.

By inspection of the force diagram of figure

1.30, a relationship between induced drag, lift,

and induced angle of attack is apparent. The

induced drag coeficient, CDi, will vary directly

with the wing lift coefficient, C,, and the in-

duced angle of attack, as. The effective lift

is the vertical component of the actual lift and,

if the induced angle of attack is small, will be

essentially the same as the actual lift. The

J horizontal and vertical component of drag is

insignificant under the same conditions. By a

detailed study of the factors involved, the fol-

lowing relationships can be derived for a wing

with an elliptical lift distribution:

(1) The induced drag equation follows the

same form as applied to any other aerody-

namic force.

Di=CDigS

where

Di=induced drag, lbs.

4= :Vymic pressures; psf

=295

Cni= induced drag coefficient

S=wing area, sq. ft.

(2) The induced drag coefficient can be

derived as :

or

CD,-C, sin ai

CD&

c,P =0.318 -Jjj ( )

where

C,= lift coefficient

sin ai=natural sine of the induced angle

of attack, Eli, degrees

r=3.1416, constant

AR= wing aspect ratio

(3) The induced angle of attack can be

derived as:

a~= 18.24 & (degrees) ( )

(NOTE: the derivation of these relationships

may be found in any of the standard engi-

neering aerodynamics textbooks.)

These relationships facilitate an understanding

and appreciation of induced drag.

The induced angle of attack Eli= 18.~4$~ ( >

depends on the lift coefficient and aspect ratio.

Flight at high lift conditions such as low speed

or maneuvering flight will create high induced

angles of attack while high speed, low lift

flight will create very small induced angles .of

attack. The inference is that high lift coefli-

cients require large downwash and result in

large ,induced angles of attack. The effect of

aspect ratio is significant since a very high

aspect ratio would produce a negligible induced

angle of attack. If the aspect ratio were in-

finite, the induced angle of attack would be

zero and the aerodynamic characteristics of the

wing would be identical with the airfoil sec-

tion properties. On the other hand, if the

wing aspect ratio is low, the induced angle of

attack will be large and the low aspect ratio

airplane must operate at high angles of attack

at maximum lift. Essentially, the low aspect

ratio wing affects a relatively small mass of

air and consequently must provide a large de-

flection (downwash) to produce lift.

EFFECT OF LIFT. The induced drag co-

e&cient (

C&l CDi=0.31E - shows somewhat sim- ,I AR

ilar effects of lift coefficient and aspect ratio.

Because of the power of variation of induced drag

coefficient with lift coefficient, high lift coefli-

cients provide very high induced drag and low

lift coefficients very low induced drag. The di-

rect effect of C, can be best appreciated by assum-

ing an airplane is flying at a givenweight, alti-

tude, and airspeed. If the airplane is maneuvered

from steady level flight to a load factor of two,

hWd Jonua~ 1965

the lift coefficient is doubled and the induced

drag is four times 0.1 grsat. If the flight load

factor is changed from one to five, the induced

drag is twenty-five times as great. If all other

factors are held constant to single out this

effect, it could be stated that “induced drag

varies as the square of the lift”

Di, ’

L! Di,= L1

where

Di,= induced drag corresponding to

some original lift, L1

Di,= induced drag corresponding to

some new lift, Lp

(and q (or EAS), S, AR are constant)

This expression defines the effect of gross

weight, maneuvers, and steep turns on the

induced drag, e.g., 10 percent higher gross

weight increases induced drag 21 percent, 4G

maneuvers cause 16 times as much induced

drag, a turn with 4s0 bank requires a load

factor of 1.41 and this doubles the induced

drag.

EFFECT OF ALTITUDE. The effect of

altitude on induced drag can be appreciated by

holding all other factors constant. The gen-

eral effect of altitude is expressed by:

where

Dil= induced drag corresponding to some orig-

inal altitude density ratio, 0,

D&= induced drag corresponding to some new

altitude density ratio, q

(and L, S, AR, V are constant)

This relationship implies that induced drag

would increase with altitude, e.g., a given

airplane flying in level flight at a given TAS

at 40,000 ft. (u=O.25) would have four times

as much induced drag than when at sea level

(u= 1.00). This effect results when the lower

NAVWEPS 0040240

BASIC AERODYNAMICS

air density requires a greater deflection of the

airstream to produce the same lift. However,

if the airplane is flown at the same EAS, the

dynamic pressure will be the same and induced

drag will not vary. In this case, the TAS

would be higher at altitude to provide the

same EAS.

EFFECT OF SPEED. The general effect of

speed on induced drag is unusual since low air-

speeds are’associated with high lift coefficients

and high lift coefficients create high induced

drag coefficients. The immediate implication

is that induced drag inmaw with decreasing air

J@. If all other factors are held constant to

single out the effect of airspeed, a rearrange-

ment of the previous equations would predict

that “induced drag varies inversely as ,the

square of the airspeed.”

where

Dil= induced drag corresponding to some orig-

inal speed, Vi

Di,= induced drag corresponding to some new

speed, Vs

(and L, S, AR, ,J are constant)

Such an effect would imply that a given air-

plane in steady flight would incur one-fourth

as great an induced drag at twice as great a

speed or four times as great an induced drag at

half the original speed. This variation may

be illustrated by assuming that an airplane in

steady level flight is slowed from 300 to 150

knots. The dynamic pressure at 1% knots is

one-fourth the dynamic pressure at 300 knots

and the wing must deflect the airstream four

times as greatly to create the same lift. The

same lift force is then slanted aft four times

as greatly and the induced drag is four times

as great.

The expressed variation of induced drag with

speed points out that induced drag will be of

greatest importance at low speeds and prac-

tically insignificant in flight at high dynamic

pressures. For example, a typical single en-

gine jet airplane at low altitude and maximum

level flight airspeed has an induced drag which

is less than 1 pcrccont of the total drag. How-

ever, this same airplane in steady flight just

above the stall speed could have an induced

drag which is approximately 75 pnrcnt of the

total drag.

EFFECT OF ASPECT RATIO. The effect

of aspect ratio on the induced drag

is the principal effect of the wing planform.

The relationship for induced drag coefIicient

emphasizes the need of a high aspect ratio

for the airplane which is continually

operated at high lift coefficients. In other

words, airplane configurations designed to

operate at high lift coefficients during the

major portion of their flight (sailplanes, cargo,

transport; patrol, and antisubmarine types)

demand a high aspect ratio wing to minimize

the induced drag. While the high aspect

ratio wing will minimize induced drag, long,

thin wings increase structural weight and have

relatively poor stiffness characteristics. This

fact will temper the preference for a very high

aspect ratio. Airplane configurations which

are developed for very high speed flight (es-

specially supersonic flight) operate at relatively

low lift coefficients and demand great aero-

dynamic cleanness. These configurations of

airplanes do not have the same preference for

high aspect ratio as the airplanes which op-

erate continually at high lift coefficients.

This usually results in the development of low

aspect ratio planforms for these airplane con-

figurations.

The effect of aspect ratio on the lift and drag

characteristics is shown in figure 1.31 for

wings of a basic 9 percent symmetrical section.

The basic airfoil section properties are shown

on these curves and these properties would be

NAVWEPS OD-SOT-BO

BASIC AERODYNAMICS

typical only of a wing planform of extremely

high (infinite) aspect ratio. When a wing of

some finite aspect ratio is constructed of this

basic section, the principal differences will be

in the lift and drag characteristics-the mo-

ment characteristics remain essentially the

same. The effect of decreasing aspect ratio on

the lift curve is to increase the wing angle of

attack necessary to produce a given lift co-

efficient. The difference between the wing

angle of attack and the section angle of attack

is the induced angle of attack, orit18.24 L AR’

which increases with decreasing aspect ratio.

The wing with the lower aspect ratio is less

sensitive to changes in angle of attack and re-

quires higher angles of attack for maximum

lift. When the aspect ratio is very low (below

3 or 6) the induced angles of attack are not

accurately predicted by the elementary equa-

tion for 01~ and the graph of C, versus 01 develops

distinct curvature. This effect is especially

true at high lift coefhcients where the lift

curve for the very low aspect ratio wing is

very shallow and CL- and stall angle of attack

are less sharply defined.

The effect of aspect ratio on wing drag char-.

acteristics may be appreciated from inspection of

figure 1.31. The basic section properties are

shown as the drag characteristics of an infinite

aspect ratio wing. When a planform of some

finite aspect ratio is constructed, the wing drag

coefficient is the rtlm of the induced drag coe&-

c,” cient, C,,=O.318 AR, and the section drag co-

efhcient. Decreasing aspect ratio increases the

wing drag coefficient at any lift coefficient since

the induced drag coefficient varies inversely

with aspect ratio. When the aspect ratio is

very low, the induced drag varies greatly with

lift and at high lift coefficients, the induced

drag is very high and increases very rapidly

with lift coefficient.

While the effect of aspect ratio on lift curve

slope and drag due to lift is an important re-

lationship, it must be realized that design for

NAVWEPS 00-8OT-80

BASIC AERODYNAMICS

0’

I--

::

t (NO SWEEPBACK)

(3

I .4 BASIC SECTION

1 \A”=‘NFl~;~lB

WING ANGLE OF ATTACK

a DEGREES

AR,=5 AR = 2.5

I I I (LOW MACH NUMBER) I

.I0 .I5 .20 .25

I WING DRAG COEFFICIENT, CD

Figure 1.31. Effect of Aspect Ratio on Wing Characteristics

NAWEPS OO-BOT-BO

BASIC AERODYNAMICS

takeoff distance may occur. Also, the initial

climb performance may be marginal at an

excessively low airspeed. There are modern

configurations of airplanes of very low aspect

ratio (plus sweepback) which-if over-

rotated during a high altitude, high gross

weight takeoff-cannot fly out of ground

effect. With the more conventional airplane

configuration, an excess angle of attack pro-

duces a well defined stall. However, the

modern airplane configuration at an excessive

angle of attack has no sharply defined stall

but developes an excessive amount of induced

drag. To be sure that it will not go unsaid,

an excessively low angle of attack on takeoff

creates its own problems-excess takeoff

speed and distance and critical tire loads.

(2) During appra& where the pilot must

exercise proper technique to control the

flight path. “Attitude plus power equals

performance.” The modern high speed con-

figuration at low speeds will have low lift-

drag ratios due to the high induced drag 1

and can require relatively high power set-

tings during the power approach. If the

pilot interprets that his airplane is below

the desired glide path, his first reaction rnu~t

trot be to just ease the nose up. An increase

in angle of attack without an increase in

power will lower the airspeed and greatly

increase the induced drag. Such a reaction

could create a high rate of descent and lead

to very undesirable consequences. The an-

gle of attack indicator coupled with the

mirror landing system provides reference to

the pilot and emphasizes that during the

steady approach “angle of attack is the

primary control of airspeed and power is the

primary control of rate of climb or descent.”

Steep turns during approach at low airspeed

are always undesirable in any type of air-

plane because of the increased stall speed and

induced drag. Steep turns at low airspeeds

in a low aspect ratio airplane can create

extremely high induced drag and can incur

dangerous sink rates.

very high speed flight does not favor the use of

high aspect ratio planforms. Low aspect ratio

planforms have structural advantages and

allow the use of thin, low drag sections for high

speed flight. The aerodynamics of transonic

and supersonic flight also favor short span, low

aspect ratio surfaces. Thus, the modern con-

figuration of airplane designed for high speed

flight will have a low aspect ratio planform

with characteristic aspect ratios of two to four.

The most important impression that should

result is that the typical modern configuration

will have high angles of attack for maximum

lift and very prodigious drag due to lift at low

flight speeds. This fact is of importance to

theNaval Aviator because the majority of pilot-

caused accidents occur during this regime of

flight-during takeoff, approach, and landing.

Induced drag predominates in these regimes of

flight.

The modern configuration of high speed air-

plane usually has a low aspect ratio planform

with high wing loading. When wing sweep-

back is coupled with low aspect ratio, the wing

lift curve has distinct curvature and is very flat

at high angles of attack, i.e., at high CL, C, in-

creases very slowly with an increase in 01. In

addition, the drag curve shows extremely rapid

rise at high lift coefficients since the drag due

to lift is so very large. These effects produce

flying qualities which are distinctly different

from a more “conventional” high aspect ratio

airplane configuration.

Some of the most important ramifications of

the modern high speed configuration are:

(1) During takeoff where the airplane must

not be over-rotated to an excessive angle of

attack. Any given airplane will have some

fixed angle of attack (and CJ which produces

the best takeoff performance and this angle

of attack will not vary with weight, density

altitude, or temperature. An excessive angle

of attack produces additional induced drag

and may have an undesirable effect on takeoff

performance. Takeoff acceleration may be

seriously reduced and a large increase in

Revised January 1965

NAVWEPS 004OT-80

BASIC AERODYNAMICS

(3) During the landing phase where an

excessive angle of attack (or excessively low

airspeed) would create high induced drag

and a high power setting to control rate of

descent. A common error in the technique

of landing modern conbgurations is a steep,

low power approach to landing. The steep

flight path requires considerable maneuver

to flare the airplane for touchdown and

necessitates a definite increase in angle of

attack. Since the maneuver of the flare is a

transient condition, the variation of both

lift and drag with angle of attack must be

considered. The lift and drag curves for a

high aspect ratio wing (fig. 1.31) show con-

tinued strong increase in C, with 01 up to stall

and large changes in Co only at the point of

stall. These characteristics imply that the

high aspect ratio airplane is usually capable

of flare without unusual results. The in-

__^_“^ :- ---I.. -c _&-__ 1. .-* n.-. -..- : 1 C,LaLDC 111 a,l5~~ VI ~LL~CL do *we p~ovmes the

increase in lift to change the flight path

direction without large changes in drag to

decelerate the airplane.

The lift and drag curves for a low aspect

ratio wing (fig. 1.31) show that at high angles

of attack the lift curve is shallow, i.e., small

changes in C, with increased a. This implies

a large rotation needed to provide the lift to

flare the airplane from a steep approach. The

drag curve for the low aspect ratio wing shows

large, powerful increases in C, with Cr. well

below the stall. These lift and drag charac-

teristics of the low aspect ratio wing create

a distinct change in the flare characteristics.

If a flare is attempted from a steep approach at

low airspeed, the increased angle of attack

may provide such increased induced drag and

rapid loss of airspeed that the airplane does not

actually flare. A possible result is that an

even higher sink rate may be incurred. This

is one factor favoring the use of the “no-flare”

or “minimum flare” type landing technique

for certain modern configurations. These same

aerodynamic properties set the best glide

speeds of low aspect ratio airplanes above the

speed for (L/D)-. The additional speed pro-

vides a more favorable margin of flare capabil-

ity for flameout landing from a steep glide path

(low aspect ratio, low (L/D)-, low glide

ratio).

The landing technique must emphasize

proper control of angle of attack and rate of

descent to prevent high sink rates and hard

landings. As before, to be sure that it will

not go unsaid, excessive airspeed at landing

creates its own problems-excessive wear and

tear on tires and brakes, excessive landing

distance, etc.

The effect of the low aspect ratio planform

of modern airplanes emphasizes the need for

proper flying techniques at low airspeeds.

Excessive angles of attack create enormous

induced drag which can hinder takeoff per-

formance and incur high sink rates at landing.

Since such aircraft have intrinsic high mini-

mum flying speeds, an excessively low angle of

attack at takeoff or landing creates its own

problems. These facts underscore the im-

portance of a “thread-the-needle,” professional

flying technique.

EFFECT OF TAPER AND SWEEPBACK

The aspect ratio of a wing is the primary

factor in determining the three-dimensional

characteristics of the ordinary wing and its

drag due to lift. However, certain local effects

take place throughout the span of the wing and

these effects are due to the distribution of area

throughout the span. The distribution of lift

along the span of a wing cannot have sharp

discontinuities. (Nature just doesn’t arrange

natural forces with sharp discontinuities.)

The typical lift distribution is arranged in

some elliptical fashion. A representative dis-

tribution of the lift per foot of span along the

scan of a wing is shown in figure 1.32.

The natural distribution of lift along the

span of a wing provides a basis for appreciating

the effect of area distribution and taper along

the span. If the elliptical lift distribution is

NAVWEPS OfJ-RDT-8D

BASIC AERODYNAMICS

A I I

TYPlChL L&i. PER kT. OF ‘SPAN ’

LIFT DISTRIBUTION

Figure 4.32. Sponwise Lift Distribution

NAVWEPS OD-8OT-80

BASIC AERODYNAMICS

matched with a planformwhose chord is dis-

tributed in an elliptical fashion (the elliptical

wing), each square foot of area along the span

produces exactly the same lift pressure. The

elliptical wing planform then has each section

of the wing working at exactly the same local

lift coefhcient and the induced downflow at

the wing is uniform throughout the span. In

the aerodynamic sense, the elliptical. wing is

the most efficient planform because the uni-

formity of lift coefficient and downwash incurs

rbt iea$t induced drag for a given aspect ratio.

The merit of any wing @anform is then meas-

ured by the closeness with which the distribu-

tion of lift coefficient and downwash approach

that of the elliptical planform.

The effect of the elliptical planform is illus-

trated in figure 1.32 by the plot of local lift

coefficient to wing lift coefficient, f! G’ versus

scm:spnn L.“CY.ICG. ,4;..t,or, Tbac e!liptical wing p*

duces a constant value of$=J.O throughout

the span from root to tip.‘ Thus, the local

section angle of attack, LYE, and local induced

angle of attack, CY,, are constant throughout

the span. If the planform area distribution is

anything other than elliptical, it may be ex-

pected that the local section and induced angles

of attack will not be constant along the span.

A planform previously considered is the

simple rectangular wing which has a taper

ratio of 1.0. A characteristic of the rectangular

wing is a strong vortex at the tip with local

downwash behind the wing which is high at

the tip and low at the root. This large non-

uniformity in downwash causes similar varia-

tion in the local induced angles of attack along

the span. At the tip, where high downwash

exists, the local induced angle of attack is

greater than the average for the wing. Since

the wing angle of attack is composed of the

sum of at and aor a large local (x, reduces the

local a0 creating low local lift coefficients at

the tip. ‘Ihe reverse is true at the root of the

rectangular wing where low local downwash

exists. This situation creates an induced angle

of attack at the root which is less than the

average for the wing and a local section angle

of attack higher than the average for the wing.

The result is shown by the graph of figure 1.32

which depicts a local lift coefficient at the root

almost 20 percent greater than the wing lift

coefficient.

The effect of the rectangular planform may

be appreciated by matching a near elliptical

lift distribution with a planform with a

constant chord. The chords near ‘the tip

develop less lift pressure than the root and

consequently have lower section lift coe&-

cients. The great nonuniformity of local lift

coefficient along the span implies that some

sections carry .more than their share of the

load while others carry less than their share

of the load. Hence, for a given aspect ratio,

the rectangular planform will be less efficient

-t-- -L. -11:. -!-~I LlLill UK C‘lqJLlCal wing. For exampie, a

rectangular wing of AR=6 would have 16

percent higher induced angle of attack for the

wing and 5 percent higher induced drag than

an elliptical wing of the same aspect ratio.

At the other extreme of taper is the pointed

wing which has a taper ratio of zero. The

extremely small parcel of area at the pointed

tip is not capable of holding the main tip

vortex at the tip and a drastic change in down-

wash distribution results. The pointed wing

has greatest downwash at the root and this

downwash decreases toward the tip. In the

immediate vicinity of the pointed tip, an

upwash is encountered which indicates that

negative induced angles of attack exist in this

area. The resulting variation of local lift

coefficient shows low cr at the root and very

high c, at the tip. This effect may be appre-

ciated by realizing that the wide chords at

the root produce low lift pressures while the

very narrow chords toward the tip are sub-

ject to very high lift pressures.. The varia-

tion of 2 throughout the span of the wing of L

taper ratio==0 is shown on the graph of figure

1.32. As with the rectangular wing, the non-

uniformity of downwash and lift distribution

result in inefficiency of rhis planform. For

example, a pointed wing of AR=6 would have

17 percent higher induced angle of attack for

the wing and 13 percent higher induced drag

than an elliptical wing of thesame aspect ratio.

Between the two extremes of taper will

exist planforms of more tolerable efficiency.

The variations of 2 for a wing of taper ratio

=0.5 closely approxtmates the lift distribution

of the elliptical wing and the drag due to lift

characteristics are nearly identical. A wing

of AR=6 and taper ratio=0.5 has only 3

percent higher ai and 1 percent greater CD: than

an elliptical wing of the same aspect ratio.

,A separate effect on the spanwise lift dis-

tribution is contributed by wing sweepback.

Sweepback of the planform tends to alter the

lift distribution similar to decreasing the taper

ratio. Also, large sweepback tends to increase

induced drag.

The elliptical wing is the ideal of the sub-

sonic aerodynamic planform since it provides

a minimum of induced drag for a given aspect

ratio. However, the major objection to the

elliptical planform is the extreme difficulty of

mechanical layout and construction. A highly

tapered planform is desirable from the stand-

point of structural weight and stiffness and

the usual wing planform may have a taper

ratio from 0.45 to 0.20. Since structural con-

siderations are quite important in the develop-

ment of an airplane configuration, the tapered

planform is a necessity for an efficient configu-

ration. In order to preserve the aerodynamic

efficiency, the resulting planform is tailored

by wing twist and section variation to obtain

as near as possible the elliptic lift distribution.

STALL PATTERNS

An additional effect of the planfotm area

distribution is on stall pattern of wing. The

desirable stall pattern of any wing is a stall

which begins on the root sections first. The

NAVWEPS OD-ROT-RO

RASIC AERODYNAM!CS

advantages of root stall first are that ailerons

remain effective at high angles of attack,

favorable stall warning results from the buffet

on the empennage and aft portion of the fuse-

lage, and the loss of downwash behind the root

usually ptovides a stable nose down moment

to the airplane. Such a stall pattern is favored

but may be difficult to obtain with certain wing

configurations. The types of stall patterns in-

herent with various planforms are illustrated

in figure 1.33. The various planform effects

are separated as follows :

(A) The elliptical planform has constant

local lift coefficients throughout the span from

root to tip. Such a lift distribution means that

all sections will reach stall at essentially the

same wing angle of attack and stall will begin

and progress uniformly throughout the span.

While the elliptical wing would reach high

lift coefficients before incipient stall, there

would be little advance warning of complete

stall. Also, the ailerons may lack effectiveness

when the wing operates near the stall and lat-

eral control may be difficult.

(B) The lift distribution of the rectangular

wing exhibits low local lift coefficients at the

tip and high local lift coe5cients at the root.

Since the wing will initiate stall in the area of

highest local lift coefficients, the rectangular

wing is characterized by a strong root stall

tendency. Of course, this stall pattern is fav-

orable since there is adequate stall warning

buffet, adequate aileron effectiveness, and usu-

ally strong stable moment changes on the ait-

plane. Because of the great aerodynamic and

structural ine&ciency of this planform, the

rectangular wing finds limited application only

to low cost, low speed light planes. The sim-

plicity of construction and favorable stall

characteristics are predominating requirements

of such an airplane. The stall sequence fot a

rectangular wing is shown by the tuft-grid

pictures. The progressive flow separation il-

lustrates the strong root stall tendency.

(C) The wing of moderate taper (taper

ratio=0.5) has a lift distribution which closely

NAVWEPS 00-SOT-80

BASIC AERODYNAMICS

.5-

SPANWISE LIFT

DISTRIBUTION

ROOT

TIP

ELLIPTICAL RECTANGULAR, X=1.0

n ~PROGRE,,,s=

MODERATE TAPER, A= 0.5 HIGH TAPER, A=O.25

Revised January 1965

Figure 1.33. Stall Patterns (sheet I of 8)

NAVWEPS OeBOT-80

BASIC AERODYNAMICS

DOWNWASH FLOW FIELD BEHIND A RECTANGULAR

WING ILLUSTRATED BY TUFT-GRID PHOTOGRAPHS

AR=2.31, k=l.O

-II- 30° OF FLOW ANGULARITY

OT

(DEG)

‘8 -

STALL

(a) TUFT GRID 6 INCHES FROM (b) TUFT GRID 24 INCHES FROM

TRAILING EDGE TRAILING EDGE

FROM NACA TN 2674

F;gure 1.33. Stall Patterns (sheet 2 of 8)

NAWEPS oD-80~~0

BASIC AERODYNAMICS

SURFACE TUFT PHOTOGRAPHS

FOR RECTANGULAR WING

AR=2.31, k-l.0

STALL

FROM NACA TN 2674

Figuse 1.33. Stall Patterns (sheet 3 of 8)

NAVWEPS Oo-8OT-80

BASIC AERODYNAMICS

DOWNWASH FLOW FIELD 8EHlNO A SWEPT TAPERED

WING ILLUSTRATED BY TUFT-GRID PHOTOGRAPHS

45’ DELTA, AR=4.0,X=O

-It-

30° OF FLOW ANGULARITY

(DEG)

STALL

STALL

(a) TUFT GRID 6 INCHES FROM (b) TUFT GRID 24 INCHES FROM

TRAILING EDGE TRALLING EDGE

FROM NACA TN 2674

Figure 1.33. Staff Patterns (sheet 4 of 81

NAVWEPS 00-BOT-BO

BASIC AERODYMAAlllCS

SURFACE TUFT PHOTOGRAPHS

FOR A SWEPT, TAPERED WING

45O DELTA, AR=4.0. x=0

i =0 DEGREES

a = 12 DEGREES

a = 8 DEGREES

B = 16 DEGREES

a = 20 DEGREES

FROM NACA TN 2674

Figure 1.33. Stall Patterns (sheet 5 of 8’)

NAVWEPS OO-SOT-80

)YE STREAMERS OhI F!ilJ MOnFl

Ftgure 7.33. Staff Patterns (sheet 6 of 8)

NAVWEPS 00-8OT-80

BASIC AERODYNAMICS

DOWNWASH FLOW FIELD BEHIND A SWEPT,TAPERED

WING ILLUSTRATED BY TUFT-GRID PHOTOGRAPHS

60° DELTA, AR=2.31, X = 0

--+-- 30” OF FLOW ANGULARITY

QT

(DEG)

STALL

STALL

(a) TUFT GRID 6 INCHES FROM (b) TUFT GRID 24 INCHES FROM

TRAILING EDGE TRAILING EDGE

FROM NACA TN 2674

Figure 1.33. Stall Patterns (sheet 7 of 8)

NAVWEPS OD-801-80

BASIC AERODYNAMICS

SURFACE TUFT PHOTOGRAHS FOR

A SWEPT, TAPERED WlNG

60° DELTA, AR=2.31, A=0

a = 0 DEGEES

/STALL

.

a =32 DEGREES

FROM NACA TN 2674

Figure 1.33. Std Patterns (sheet 8 018)

NAVWEPS 00-801-80

BASIC AERODYNAMICS

approximates that of the elliptical wing.

Hence, the stall pattern is much the same as the

elliptical wing.

(D) The highly tapered wing of taper

ratio=0.25 shows the stall tendency inherent

with high taper. The lift distribution of such

a wing has distinct peaks just inboard from the

tip. Since the wing stall is started in the

vicinity of the highest local lift coefficient,

this planform has a strong “tip stall” tendency.

The initial stall is not started at the exact tip

but at the station inboard from the tip where

highest local lift c,oefficients prevail. If an

actual wing were allowed to stall in this

fashion the occurrence of stall would be typi-

fied by aileron buffet and wing drop. There

would be no buffet of the empennage or aft

fuselage, no strong nose down moment, and

very little-if any-aileron effectiveness. In

order to prevent such undesirable happenings,

the wing must be tailored to favor the stall

pattern. The wing may be given a geometric.

twist or “washout” to decrease the local

angles of attack at the tip. In addition, the

airfoil section may be varied throughout the

span such that sections with greater thickness

and camber are located in the areas of highest

local lift coefhcients. The higher ct- of

such sections can then develop the higher local

C~S and be less likely to stall. The addition

of leading edge slots or slats toward the tip

increase the local c t- and stall angle of attack

and are useful in allaying tip stall and loss of

aileron effectiveness. Another device for im-

proving the stall pattern would be the forcing

of stall in the desired location by decrctising the

section ctmar in this vicinity. The use of sharp

leading edges or “stall strips” is a powerful

device to control the stall pattern.

.(E) The pointed tip wing of taper ratio

equal to zero develops extremely high local

lift coefficients at the tip. For all practical

purposes, the pointed tip will be stalled at any

condition of lift unless extensive tailoring is

applied to the wing. Such a planform has no

practical application to an airplane which is

definitely subsonic in performance.

(F) Sweepback applied to a wing planform

alters the lift distribution similar to decreasing

taper ratio. Also, a predominating influence

of the swept planform is the tendency for a

strong crossflow of the boundary layer at high

lift coefficients. Since the outboard sections

of the wing trail the inboard sections, the out-

board suction pressures tend to draw the

boundary layer toward the tip. The result is

a thickened low energy boundary layer at the

tips which is easily separated. The develop

ment of the spanwise flow in the boundary

layer is illustrated by the photographs of

figure 1.33. Note that the dye streamers on

the upper surface of the~swept wing develop a

strong spanwise crossflow at high angles of

attack. Slots, slats, and flow fences help to

allay the strong tendency for spanwise flow.

When sweepback and taper are combined in

a planform, the inherent tip stall tendency is

considerable. If tip stall of any significance is

allowed to occur on the swept wing, an addi-

tional complication results: the forward shift

in the wing center of pressure creates an un-

stable nose up pitching moment. The stall

sequence of a swept, tapered wing is indicated

by the tuft-grid photographs of figure 1.33.

An additional effect on sweepback is the re-

duction in the slope of the lift curve and maxi-

mum lift coeflicient. When the sweepback is

large and combined with low aspect ratio the

lift curve is very shallow and maximum lift

coefficient can occur at tremendous angles.of

attack. The lift curve of one typical low

aspect ratio, highly tapered, swept wing air-

plane depicts a maximum lift coefficient at

approximately 43’ angle of attack. Such dras-

tic angles of attack are impractical in many

respects. If the airplane is operated at such

high angles of attack an extreme landing gear

configuration is required, induced drag is ex-

tremely high, and the stability of the airplane

may seriously deteriorate. Thus, the modern

conhguration of airplane may have “minimum

control speeds” set by these factors rather than

simple stall speeds based on C&,.

When a wing of a given planform has various

high lift devices added, the lift distribution and

stall pattern can be greatly affected. Deflec-

tion of trailing edge flaps increases the local

lift coe5cients in the flapped areas and since

the stall angle of the flapped section is de-

creased, initial stall usually begins in the

flapped area. The extension of slats simply

allows the slatted areas to go to higher lift

coe5cients and angles of attack and generally

delays stall in that vicinity. Also, power

effects may adversely affect the stall pattern of

the propeller powered airplane. When the

propeller powered airplane is at high power

and low speed, the flow induced at the wing

root by the slipstream may cause considerable

delay in the stall of the root sections. Hence,

the propeller powered airplane may have its

most undesirable stall characteristics during the

power-on stall rather than the power-off stall.

PARASITE DRAG

In addition to the drag caused by the de-

velopment of lift (induced drag) there is the

obvious drag which is nor due to the develop

ment of lift. A wing surface even at zero lift

will have “profile” drag due to skin friction

and form. The other components of the air-

plane such as the fuselage, tail, nacelles, etc.,

contribute to drag because of their own form

and skin friction. Any loss of momentum of

the airstream due to powerplant cooling, air

conditioning, or leakage through construction

or access gaps is, in effect, an additional drag.

When the various components of the airplane

are put together the total drag will be greater

than the sum of the individual components

because of “interference” of one surface on the

other.

The most usual interference of importance

occurs at the wing-body intersection where the

growth of boundary layer on the fuselage re-

duces the boundary layer velocities on the wing

root surface. This reduction in energy allows

NAVWEPS OO-ROl-80

BASIC AERODYNAMICS

the wing root boundary layer to be more easily

separated in the presence of an adverse pressure

gradient. Since the upper wing surface has the

more critical pressure gradients, a low wing

position on a circular fuselage would create

greater interference drag than a high wing

position. Adequate filleting and control of

local pressure gradients is necessary to mini-

mize such additional drag due to interference.

The sum of all the drags due to form, fric-

tion, leakage and momentum losses, and inter-

ference drag is termed “parasite” drag since

it is not directly associated with the develop-

ment of lift. While this parasite drag is not

directly associated with the production of lift

it is a variable with lift. The variation of

parasite drag coefficient, C+, with lift coef-

ficient, C,, is shown for a typical airplane in

figure 1.34. The minimum parasite drag co-

efficient, CDpmi,, usually occurs at or near zero

lift and parasite drag coefficient increases

above this point,in a smooth curve. The in-

duced drag coefficient is shown on the same

graph for purposes of comparison since the

total drag of the airplane is a sum of the

parasite and induced drag.

In many parts of airplane performance it is

necessary to completely distinguish between

drag due to lift and drag not due to lift. The

total drag of an airplane is the sum of the para-

site and induced drags.

G=c++cD;

where

C, = airplane drag coefficient

C+=parasite drag coefficient

C,,= induced drag coeaicient

From inspection of figure 1.34 it is seen that

both CD, and CD, vary with lift coefticient.

However, the usual variation of parasite drag

allows a simple correlation with the induced

drag term. In effect, the part of parasite drag

above the minimum at zero lift can be “lumped”

NAVWEPS 00-801-80

BASIC AERODYNAMICS

1.4

1.2

i 0.4

0.2

0 .05 ;!O .!5

DRAG COEFFICIENT, CD

I.4

I.2

j 1.0

^

t 0.6

kl $ 0.6

i 0.4

0.2

DRAG COEFFICIENT, CD

Figure 1.34. Airplane Parasite and Induced Drag

NAVWEPS 00-8OT-80

BASIC AERODYNAMICS

ure is not too accurate because of the sharper

variation of parasite drag at high angles of

attack. In a sense, the airplane efficiency fac-

tor would change from the constant value and

decrease. The deviation of the actual airplane

drag from the approximating curve is quite

noticeable for airplanes with low aspect ratio

and sweepback. Another factor to consider is

the effect of compressibility. Since compressi-

bility effects would destroy this relationship,

the greatest application is for subsonic perform-

ance analysis.

The total airplane drag is the sum of the

parasite and induced drags.

where

D= D,+D<

Di= induced drag

in with the induced drag coefficient by a con-

stant factor which is defined as the “airplane

e5ciency factor”, c. By this method of ac-

counting the airplane drag coe5cient is ex-

pressed as :

where

C DPmB=

minimum parasite drag

coefficient

CD;= induced drag coe5cient

e = airplane e5ciency factor

In this form, the airplane drag coefficient is

expressed as the sum of drag not due to lift

F%d” ) and drag due to lift (G). The air-

plane efficiency factor is some co&ant (usually

less than unity) which includes parasite drag

due to lift with the drag induced by lift.

C Dpmr” is invariant with lift and represents the

parasite drag at zero lift. A typical value of

C r,Pmin would be 0.020, of which the wing may

account for 50 percent, the fuselage and nacelles

40 percent, and the tail 10 percent. The term

of ( 0.318 g > accounts for all drag due’ to

lift-the drag induced by lift and the extra

parasite drag due to lift. Typical values of

the airplane efficiency factor range from 0.6 to

0.9 depending on the airplane configuration

and its characteristics. While the term of

drag due to lift does include some parasite

drag, it is still generally referred to as induced

drag.

The second graph of figure 1.34 shows that

the sum of CD, and G can approximate the -mm e

actual airplane CD through a large range of lift

coefficients. For airplanes of moderate aspect

ratio, this representation of the airplane total

drag is quite accurate in the ordinary range of

lift coefficients up to near 70 percent of CL,,.

At high lift coefficients near CL-, the proced-

and

=(0.318 $+S

D,= parasite drag

When expressed in this form the induced drag,

Di, includes all drags due to lift and is solely

a function of lift. The parasite drag, D,, is

the parasite drag and is completely independent

of lift-it could be called the “barn door”

drag of the airplane.

An alternate expression for the parasite drag

is:

R=fq

where

f = equivalent parasite area, sq. ft.

f = CDPmi,S

q= dynamic pressure, psf

UP =-

or

DpEfg

In this form, the equivalent parasite area, f,

is the product of CDPml” and S and relates an

B I Y

impression of the “barn door” size. Hence,

parasite drag can be appreciated as the result

of the dynamic pressure, 4, acting on the

equivalent parasite area, j. The “equivalent”

parasite area is defmed by this relationship as

a hypothetical surface with a C,=l.O which

produces the same parasite drag as the air-

plane. An analogy would be a barn door in

the airstream which is equivalent to the air-

plane. Typical values for the equivalent para-

site area range from 4 sq. ft. for a clean fighter

type airplane to 40 sq. ft. for a large transport

type airplane. Of course, when any airplane

is changed from the clean configuration to the

landing configuration, the equivalent parasite

area increases.

EFFECT OF CONFIGURATION. The par-

asite drag, D,, is unaffected by lift, but is

variable with dynamic pressure and equivalent

parasite area. This principle furnishes the

basis for illustrating the variation of parasite

drag with the various conditions of flight.

If all other factors are held constant, the para-

site drag varies directly with the equivalent

parasite area.

D,,= b C) D,, I

where

D,,= parasite drag corresponding to some orig-

inal parasite area, fi

D,,==parasite drag corresponding to some new

parasite area, fi

(V and (r are constant)

As an example, the lowering of the landing

gear and flaps may increase the parasite area

80 percent. At any given speed and altitude

this airplane would experience an 80 percent

increase in parasite drag.

EFFECT OF ALTITUDE. In a similar man-

ner the effect of altitude on parasite drag may

NAVWEK OD-BOT-BO

BASIC AERODYNAMICS

be appreciated. The general effect of altitude

is expressed by:

where

D,, = parasite drag corresponding to some orig-

inal altitude density ratio, 0,

D,,=parasite drag corresponding to some new

altitude density ratio, (ra

(and f, V are constant)

This relationship implies that parasite drag

would decrease at altitude, e.g., a given air-

plane in flight at a given T.4.Y at 40,COO ft.

(e=O.29 would have one-fourth the parasite

drag when at sea level (u=l.OO). This effect

results when the lower air density produces

less dynamic pressure. However, if the air-

plane is flown at a constant EAS, the dynamic

pressure and, thus, parasite drag do not vary.

In this case, the TASwould be higher at altitude

to provide the same EAS.

EFFECT OF SPEED. The effect of speed

alone on parasite drag is the most important.

If all other factors are held constant, the effect

of velocity on parasite drag is expressed as:

&, V, * -=- (3 D,, V

where

D,,=parasite drag corresponding to some orig-

inal speed, Vi

D,,=parasite drag corresponding to some new

speed, VS

(j and o are constant)

This relationship expresses a powerful effect

of speed on parasite drag. As an example, a

given airplane in flight at some altitude would

have four times as much parasite drag at twice

NAVWEPS 00-801-80

BASIC AERODYNAMICS

as great a speed or one-fourth as much parasite

drag at half the original speed. This fact may

be appreciated by the relationship of dynamic

pressure with speed-twice as much V, four

times as much 4, and four times as much D,.

This expressed variation of parasite drag with

speed points out that parasite drag will be of

greatest importance at high speeds and prac-

tically insignificant in flight at low dynamic

pressures. To illustrate this fact, an airplane

in flight just above the stall speed could have a

parasite drag which is only 25 percent of the

total drag. However, this same airpfane at

maximum level flight speed at low altitude

would have a parasite drag which’ is very

nearly 100 percent of the total drag. The

predominance of parasite drag at high flight

speeds emphasizes the necessity for great aero-

dynamic cleanness (low j) to obtain high speed

performance.

In the subsonic regime of flight, the ordinary

configuration of airplane has a very large por-

tion of the equivalent parasite area determined

by skin friction drag. As the wing contrib-

utes nearly half of the total parasite drag, the

profile drag of the wing can be minimized by

the use of the airfoil sections which produce

extensive laminar flow. A subtle effect on

parasite drag occurs from the influence of the

wing area. Since the wing area (S) appears

directly in the parasite drag equation, a reduc-

tion in wing area would reduce the parasite

drag if all other factors were unchanged.

While the exact relationship involves con-

sideration of many factors, most optimum

airplane configurations have a strong preference

for the highest practical wing loading and

minimum wing surface area.

As the flight speeds of aircraft approach the

speed of sound, great care must be taken to

delay and alleviate compressibility effects.

In order to delay and teduce the drag rise

associated with compressibility effects, the

components of the airplanes must be arranged

to reduce the early formation of shock waves

on the airplane. This will generally require

fuselage and nacelles of high fineness ratio,

well faired canopies, and thin wing sections

which have very smooth uniform pressure dis-

tributions. -Low aspect ratios and sweepback

are favorable in delaying and reducing the

compressibility drag rise. In addition, inter-

ference effects are quite important in transonic

and supersonic flight and the airplane cross

section area distribution must be controlled

to minimize local velocity peaks which could

create premature strong shock wave formation.

The modern configuration of airplane will

illustrate the features required to effect very

high speed performance-low aspect ratio,

sweepback, thin low drag sections, etc. These

same features produce flight characteristics at

low airspeeds which necessitate .proper flying

technique.

AIRPLANE TOTAL DRAG

I%,- rn+ql Jr,, nf ~ln eimlooe in fl.jght is the AI&C CYCYl Y Ye v YIL L”y’““c

sum of the induced and parasite drag. Figure

I.35 illustrates the variation of toral drag

with speed for a given airplane in level flight

at a particular weight, configuration, and alti-

tude. The parasite drag increases with speed

varying as the square of the velocity while the

induced drag decreases with speed varying in-

versely as the square of the velocity. The

total drag of the airplane shows the predomi-

nance of induced drag at low speed and parasite

drag at high speed. Specific points of interest

on the drag curve are as follows:

(A) Stall of this particular airplane occurs

at 100 knots and is indicated by a sharp rise

in the actual drag. Since the generalized iqua-

tions for induced and parasite do not account

for conditions at stall, the actual drag of the

airplane is depicted by the “hook” of the

dotted line.

(B) At a speed of 124 knots, the airplane

would incur a minimum rate of descent in

power-off flight. Note that at this speed the

induced drag comprises 75 percent of the total

drag. If this airplane were powered with a

reciprocating-propeller type powerplant, maxi-

mum endurance would occur at this airspeed.

NAVWEPS OO-ROT-80

BASIC AERODYNAMICS

VELOCITY KNOTS

Figure 9.35. Typical Airplane Drag Curves

NAVWEPS OO-BOT-80

BASIC AE,RODYNAMlCS

(C) The point of minimum total drag occurs

at a speed of 163 knots. Since this speed in-

curs the least total drag for lift-equal-weight

flight, the airplane is operating at (L/D)ma,.

Because of the particular manner in which

parasite and induced drags vary with speed

(parasite drag directly as the speed squared;

induced drag inversely as the speed squared)

the minimum total drag occurs when the in-

duced and parasite drags are equal. The speed

for minimum drag is an important reference for

many items of airplane performance. One

item previously ,presented related glide per-

formance and lift-drag ratio. At the speed of

163 knots this airplane incurs a total drag of

778 lbs. while producing 12,000 lbs. of lift.

These figures indicate a maximum lift-drag

ratio of 15.4.and relate a glide ratio of 15.4.~

In addition, if this airplane were jet powered,

the airplane would achieve maximum en-

durance at this airspeed for ‘the specified alti-

tude. If this airplane were propeller powered,

the airplane would achieve maximum range at

this airspeed for the specified altitude.

(D) Point (D) is at an airspeed approxi-

mately 32 percent greater than the speed for

(L/D),.,. Note that the parasite drag com-

prises 75 percent of the total drag at a speed of

215 knots. This point on the drag curve pro-

duces the highest proportion between velocity

and drag and would be the point for maximum

range if the airplane were jet powered. Be-

cause of the high proportion of parasite drag

at this point the long range jet airplane has

great preference for great aerodynamic clean-

ness and less demand for a high aspect ratio

than the long range propeller powered airplane.

(E) At a speed of 400 knots, the induced

drag is an extremely small part of the total

drag and parasite drag predominates.

(P) As the airplane reaches very high flight

speeds, the drag rises in a very rapid fashion

due to compressibility. Since the generalized

equation for parasite drag does not account for

compressibility effects, the actual drag rise is

typified by the dashed line.

The airplane drag curve shown in figure 1.34

is particular to one weight, configuration, and

altitude in level flight. Any change in one of

these variables will affect the specific drags at

specific velocities.

The airplane drag curve is a major factor in

many items of airplane performance. Range,

endurance, climb, maneuver, landing, takeoff,

etc., performance are based on some relation-

ship involving the airplane drag curve.

NAVWEPS 00-8OT-80

AIRPLANE PERFORMANCE

The performance of an aircraft is. the most operating limitations and insight to obtain

important feature which defines its suitability the design performance of his aircraft. The

for specific missions. The principal items of performance section of the flight handbook

airplane performance deserve detailed consid- provides the specific information regarding the

eration in order to better understand and capabilities and limitations of each airplane.

appreciate the capabilities of each airplane.

Knowledge of the various items of airplane

Every Naval Aviator must rely upon these

handbook data as the guide to safe and effec-

performance will provide the Naval Aviator rive operation of his aircraft.

with a more complete appreciation of the

NAVWEPS 00-ROT-80

AIRPLANE PER,FORMANCE

REQUIRED THRUST AND POWER

DEFINITIONS

All of the principal items of flight perform-

ance involve steady state flight conditions and

equilibrium of the airplane. For the airplane

to remain in steady level flight, equilibrium

must be obtained by a lift equal to the air-

plane weight and a powerplant thrust equal to

the airplane drag. Thus, the airplane drag

defines the thrust required to maintain steady

level flight.

The total drag of the airplane is the sum of

the parasite and induced drags: Parasite drag

is the sum of pressure and friction drag which

is due to the basic configuration and, as de-

fined, is independent of lift. Induced drag is

the undesirable but unavoidable consequence

of the development of lift. In the process of

creating lift by the deflection of an airstream,

the actuai iift is inclined and a coimponcn: of

lift is incurred parallel to the flight path direc-

tion. This component of lift combines with

any change in pressure and friction drag due

to change in lift to form the induced drag.

While the parasite drag predominates at high

speed, induced drag predominates at low speed.

Figure 2.1 illustrates the variation with speed

of the induced, parasite, and total drag for a

specific airplane configuration in steady level

flight.

The power required for flight depends on the

thrust required and the flight velocity. By

definition, the propulsive horsepower required

is related to thrust required and flight velocity

by the following equation:

pr= Trv

3%

where

Pr=power required, h.p.

Tr= thrust required (total drag), Ibs.

V= true airspeed, knots

By inspection of this relationship, it is appar-

ent that each’pound of drag incurred at 325

knots requires one horsepower of propulsive

power. However, each pound of drag at 650

knots requires two horsepower while each

pound of drag at 162.5 knots requires one-half

horsepower. The term “power” implies work

rate and, as such, will be a function of the speed

at which a particular force is developed.

Distinction between thrust required and

pawcr required is necessary for several reasons.

For the items of performance such as range and

endurance, it is necessary to relate powerplant

fuel flow with the propulsive requirement for

steady IeveI flight. Some powerplants incur

fuel flow rate according to output thrust while

other powerplants incur fuel flow rate depend-

ing on output power. For example, the turbo-

jet engine is principally. a thrust producing

machine and fuel flow is most directly related

to thrust output. The reciprocating engine is

principally a power producing machine and

fuei flow is most directiy reiated to power

output. For these reasons the variation of

thrust required wil1 be of greatest interest in

the performance of the turbojet powered air-

plane while the variation of power required

will be of greatest interest in the performance

of the propeller powered airplane. Also, dis-

tinction between power and thrust required is

necessary in the study of climb performance.

During a steady climb, the rate of climb will

depend on excess power while the angle of

climb is a function of excess thrust.

The total power required for flight can be

considered as the sum of induced and parasite

effects similar to the total drag of the airplane.

The induced power required is a function of the

induced drag and velocity.

p,,,!g

where

Pri= induced power required, h.p.

D<=induced drag, lbs.

V= true airspeed, knots

Thus, induced power required will vary with

lift, aspect ratio, altitude, etc., in the same

manner as the induced drag. The only differ-

ence will be the variation with speed. If all

other factors remain constant, the induced

power required varies inversely with velocity

while induced’drag varies inversely with the

square of the velocity.

where

Pri,=induced power required corresponding to

some original speed, Vi

I+;,= induced power required corresponding to

some different speed, V,

For example, if an airplane in steady level flight

is operated at.twice as great a speed, the in-

duced drag is one-fourth the original value but

the induced power required is one-half the

original value.

The parasite power required is a function

of the parasite drag and velocity.

where

Pr,=parasite power required, h.p.

D,=paraSite drag, lbs.

V= true airspeed, knots

Thus, parasite power required will vary with

altitude and equivalent parasite area ( f) in the

same manner as ‘the parasite drag. However,

the variation with speed will be different. If

all other factors are constant, the parasite drag

varies as the square of velocity but parasite

power varies as the cube of velocity.

Pb% v* 3

-=(-I Ph VI

where

Prpl= parasite power required corresponding to

some original speed, Vi

NAVWEPS 00-8OT-80

AIRPLANE PERFORMAN:CE

PrPs=parasite power required corresponding to

some different speed, I’,

For example, if an airplane in steady flight is

operated at twice as great a speed, the parasite

drag is four times as great but the parasite

~;;zr required is eight times the original

Figure 2.1 presents the thrust required and

power required for a specific airplane configu-

ration and altitude. The curves of figure 2.1

are applicable for the following airplane data:

gross weight, W= 15,000 Ibs.

span, b=40 ft.

equivalent parasite area, f=7.2 sq. ft.

airplane efficiency factor, c= ,827

sea level altitude, C= 1.000

compressibility corrections neglected

The curve of drag or thrust required versus

velocity shows the variation of induced, para-

site, and total drag. Induced drag predomi-

nates at low speeds. When the airplane is

operated at maximum lift-drag ratio, (L/D)-,

the total drag is at a minimum and the induced

and parasite drags are equal. For the specific

airplane of figure 2.1, (,L/D),, and minimum

total drag are obtained at a speed of 160 knots.

The curve of power required versus velocity

shows the variation of induced, parasite, and

total power required. As before, induced

power required predominates at low speeds and

parasite power required predominates at high

speeds and the induced and parasite power are

equal at (L/D),,. However, the condition of

(L/D&- defines only the point of minimum

drag and does not define the point of minimum

pozver required. Ordinarily, the point of mini-

mum power required will occur at a speed

which is 76 percent of the speed for minimum

drag and, in the case of the airplane configura-

tion of figure 2.1, the speed for minimum power

required would be 122 knots. The total drag

at the speed for minimum power required is 15

percent higher than the drag at (L/D)- but the

minimum power required is 12 percent lower

than the power required at (L/D)-.

NAVWEPS OO-ROT-80

AIRPLANE PERFORMANCE

Figure 2.1. Airplane Thrust and Power Required

NAVWEPS OO-.ROT-80

AtRPlANE PERFORMANCE

Induced drag predominates at speeds below

the point of minimum total drag. When the

airplane is operated at the condition of mini-

mum power required, the total drag is 75

percent induced drag and 25 percent parasite

drag. Thus, the induced drag is three times as

great as the parasite drag when at minimum

power required.

VARIATIONS OF THRUST REQUIRED AND

POWER REQUIRED

The curves of thrust required and power

required versus velocity provide the basis for

comprehensive analysis of all the major items

of airplane performance. The changes in the

drag and power curves with variations of air--

plane gross weight, configuration, and altitude

furnish insight for the ‘variation of range,

endurance, climb performance, etc., with these

same items.

The effect of a change in weight on the thrust

and power required is illustrated by figure 2.2.

1 The primary effect of a weight change is a

change in the induced drag and induced power

required at any given speed. Thus, the great-

est changes in the curves of thrust and power

required will take place in the range of low

speed flight where the induced effects pre-

dominate. The changes in thrust and power

required in the range of high speed flight are

relatively slight because parasite effects pre-

dominate at high speed. The induced effects

at high speed are relatively small and changes

in these items produce a small effect on the

total thrust or power required.

In addition to the general effect on .the in-

duced drag and power required at particular

speeds, a change in weight will require that the

airplane operate at different airspeeds to main-

tain conditions of a specific lift coefficient and

angle of attack. If the airplane is in steady

flight at a particular C,,, the airpseed required

for this CL will vary with weight in the fol-

lowing manner :

v, Tg -=J VI E

where

Vi = speed corresponding to a specific C,

and weight, W,

Va=speed corresponding to the same C,

but a different weight, Ws

For the example airplane of figure 2.2, a change

of gross weight from 15,000 to 22,500 lbs. re-

quires that the airplane operate at speeds which

are 22.5 percent greater to maintain a specific

lift coefficient. For example, if the 15,000-lb.

airplane operates at 160 knots for (L/D)-, the

speed for (L/D)mz at 22,500 lbs. is:

v, = VI@

=I&) 22,500

-\i- 15,000

= (160) (1.225)

= 196 knots

The same situation exists with respect to the

curves of power required where a change in

weight requires a change of speed to maintain

flight at a particular CL. For example, if the

15,000-lb. airplane achieves minimum power

required at 122 knots, an increase in weight to

22,500 Ibs. increases the speed for minimum

power required to 149 knots.

0f course, the thrust and power required at

specific lift coefficients are altered by changes in

weight. At a specific C,, any change in weight

causes a like change in thrust required, e.g., a

50-percent increase in weight causes a 50-per-

cent increase in thrust required at the same C,.

The effect of a weight change on the power re-

quired at a specific CL is a bit more complex be-

cause a change in speed accompanies the change

Revised January 1965

NAVWEPS OO-ROT-80

AIRPLANE PERFORMANCE

Figure 2.2. Effect of Weight on Thrust and Power Required

in drag and there is a two-fold effect. A 50-

percent increase in weight produces an increase

of 83.8 percent in the power required to main-

tain a specific CL. This is the result of a 50-

percent increase in thrust required coupled with

a 22.5-percent increase in speed. The effect of a

weight change on thrust required, power re-

quired, and airspeed at specific angles of attack

and lift coefficients provides an important basis

for various techniques of cruise and endurance

conditions of flight.

1 Figure 2.3 illustrates the effect on the curves

of thrust and power required of a change in the

equivalent parasite area,!, of the configuration.

Since parasite drag predominates in the region

of high flight speed, a change in f will produce

the greatest change in thrust and power re-

quired at high speed. Since parasite drag is

relatively small in the region of low speed

flight, a change in f will produce relatively

small changes in thrust and power required at

low speeds. The principal effect of a change in

equivalent parasite area of the configuration is

to change the parasite drag at any given air-

speed.

The curves of figure 2.3 depict the changes in

the curves of thrust and power required due

to a 50 percent increase in equivalent parasite

area of the configuration. The minimum total

drag is increased by an increase in f and the

GWL is reduced. ‘Also, the increase in f

will increase the CL for (L/D)- and require a

reduction in speed at the new, but decreased,

(L/D)-. The point of minimum power re-

quired occurs at a lower airspeed and the value

of the minimum power required is increased

slightly. Generally, the effect on the mini-

mum power required is slight because the para-

site drag is only 25 percent of the total at this

specific condition of flight.

An increase in the equivalent parasite area

of an airplane may he brought about by the

deflection of flaps, extension of landing gear,

extension of speed brakes, addition of external

stores, etc. In such instances a decrease in the

airplane efficiency factor, c, may accompany

NAVWEPS 00-501-50

AMPLANE PERFORMANCE

an increase in f to account for the additional

changes in parasite drag which may vary with

C‘.

A change in altitude can produce signifi-

cant changes in the curves of thrust and power

required. The effects of altitude on these

curves providea great part of the explanation of

the effect of altitude on range and endurance.

Figure 2.4 illustrates the effect of a change in

altitude on the curves of thrust and power re-

quired for a specific airplane configuration and

gross weight. As long as compressibility

effects are negligible, the principal effect of

increased altitude on the curve of thrust re-

quired is that specific aerodynamic conditions

occur at higher true airspeeds. For example,

the subject airplane at sea level has a minimum

drag of 1,250 lbs. at 160 knots. The same

airplane would incur the same drag at altitude

if operated at the same cqthdcnt airsprcd of 160

knots. However, the equivalent airspeed of

160 knots at 22,000 ft. altitude would produce

a true airspeed of 227 knots. Thus, an in-

crease in altitude will cause the curve of thrust

required to flatten out and move to the direc-

tion of higher velocity. Note that altitude

alone will not alter the value of minimum drag.

The effect of altitude on the curve of power

required can best be considered from the effect

on true airspeed to achieve a specific aero-

dynamic condition. The sea level power re-

quired curve of figure 2.4 indicates that

CW>mz occurs at 160 knots and requires 615

h.p. If this same airplane is operated at

WD)ma at an altitude of 22,000 ft., the same

drag is incurred at a higher velocity and re-

quires a higher power. The increase in ve-

locity to 227 knots accounts for the increase

in power required to 872 hp. Actually, the

various points on the curve of power required

can be considered affected in this same fashion.

At specific lift coefficients and angles of attack,

a change in altitude will alter the true airspeed

particular to these points and cause a change

in power required because of the change in

true airspeed. An increase in altitude will

Revised Januaty 1965

NAVWEPS 00-8OT-80

AIRPLANE PERFORMANCE

VELOCITY-KNOTS

VELOCITY-KNOTS

Figure 2.3. Effect of Equivalent Parasite Area, f, on Thrust and Power Required

NAVWEPS Oo-8oT-80

AIRPLANE PERFORMANCE

THRUST

REQUIRED

(LB9

VELOCITY-KNOTS (TAS)

POWER

REK?

:D

VELOCITY-KNOTS (TAS)

Figure 2.4. Ekf of Altitude on Thrust and Power Required

NAVWEPS 00-8OT-80

AIRPLANE PERFORMANCE

cause the power required curve to flatten out

and move to higher velocities and powers

required.

The curves of thrust and power required and

their variation with weight, altitude, and con-

figuration are the basis of all phases of airplane

performance. These curves define the require-

lnent~ of the airplane and must be considered

with the power and thrust available from the

powerplants to provide detailed study of the

various items of airplane performance.

AVAILABLE THRUST AND POWER

PRINCIPLES OF PROPULSION

All powerplants have in common certain

general principles. Regardless of the type of

propulsion device, the development of thrust is

related by Newton’s laws of motion.

or

where

F=ma

F-d(mV)

df

$=force or thrust, lbs.

m=mass, slugs

a=acceleration, ft. per sec.%

d=derivative with respect to time, e.g.,

dr rate of change with time

mV=momentum, lb.-sec., product of mass

and velocity

The force of thrust results from the accelera-

tion provided the mass of working fluid. The

magnitude of thrust is accounted for by the

rate of change of momentum produced by the

powerplant. A rocket powerplant creates

thrust by creating a very large change in veloc-

ity of a relatively small mass of propellants.

A propeller produces thrust by creating a com-

paratively small change in velocity of a rela-

tively large mass of air.

The development of thrust by a turbojet or

ramjet powerplant is illustrated by figure 2.5.

Air approaches at a velocity, Vi, depending on

the flight speed and the powerplant operates

on a certain mass flow of air, Q, which passes

through the engine. Within the powerplant

the air is compressed, energy is added by the

burning of fuel, and the mass flow is expelled

from the nozzle finally reaching a velocity,

V;. The momentum change accomplished bv

this action produces the thrust,

where

Ttz=Q (V,V,)

Ta= thrust, lbs.

Q= mass flow, slugs per sec.

Vi= inlet (or flight) velocity, ft. per sec.

V,= jet velocity, ft. per sec.

The typical ramjct or turbojet powerplane de-

rives its thrust by working with a mass flow

relatively smaller than that of a propeller but

a relatively greater change of velocity. From

the previous equation it should be appreciated

that the jet thrust varies directly with the mass

flow Q, and velocity change, Va-Vi. This

fact is useful in accounting for many of the

performance characteristics of the jet power-

plant.

In the process of creating thrust by mo-

mentum change of the airstream, a relative

velocity, Vz-V1, is imparted to the airstream.

Thus, some of the available energy is essen-

tially wasted by this addition of kinetic energy

to the airstream. The change of kinetic energy

per time can account for the power wasted in

the airstream.

Pw=KE/t