Mattingly J.D., Heiser W.H., Pratt D.T. Aircraft Engine Design

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SIZING THE ENGINE: INSTALLED PERFORMANCE 193

"additive drag"

Dadd = (P - PO) dA

Applying stream thrust analysis to the stream tube control volume between station 0

and station 1, we find that

(a, )

Dada

= P1AI(1

+ yM 2) - PoAo ~ + gMg

(6.5)

A good inlet would, of course, recover most of this force through suction on

the external surface, or "lip," of the inlet, but the accompanying adverse pressure

gradients make boundary-layer separation a constant danger, in which case the

additive drag will not be regained. An unfortunate fact is that larger additive drags

are harder to regain because they are associated with more severe adverse pressure

gradients along the cowl. Please note that extemal friction is not included in this

analysis because external viscous forces are included in airplane drag as far as

"accounting" goes.

Therefore, a reasonable "worst case" or upper limit, for subsonic inlet drag

would be to assume massive separation and no recovery of additive drag. For this

situation, Eqs. (6.2a) and (6.5), conservation of mass, and the usual perfect gas

relationships can be combined to show that

M0~(1 +

yM 2) - (A]/Ao + yM 2)

Dadd M1

~ginlet - /no( F /rho) -- { F g¢/ (rhoao)}(g Mo)

(6.6a)

which can easily be evaluated assuming adiabatic, isentropic flow for any set of

values of M0, M1, a0, and

F/rho.

The static inlet drag may be estimated from the

reduced form of Eq. (6.6a), as follows:

(1 + yM2)/~/1 +(y- 1)M12/2- [1 +(y- 1)M2/2] ~+-~'

(~inlet/static

{ F gc/(rhoao)}(y Ml)

(6.6b)

Figure 6.2a shows the results of calculations for a variety of M0 and M1 combi-

nations and

Fgc/(rhoao)

= 4. Figure 6.2b shows the static inlet drag for several

Fg¢/(~noao)

values ranging from a high of 4 (afterbuming fighter engines) to a

low of 0.5 (high bypass transport engines). The information contained there leads

to the following conclusions:

~ginle t

is not large if M0 is near M1. This is the natural result of D~aa being

exactly zero when M0 = M1 and the entering stream tube experiences neither a

change in flow area nor streamwise pressure forces on the surface. In this region,

lip separation can

still

occur, especially during vigorous maneuvers, and the drag

estimate of Eq. (6.6) may be low.

2) For the usual range of subsonic flight (0.2 < M0 < 0.9), it is desirable to keep

M1 in the vicinity of 0.4-0.6.

3) Unacceptably high values of

cbi, let

occur at M0 = 0, the beginning of every

flight, reaching at least 0.05-0.10, but as much as 0.40-0.50. This explains the

0.05

0.04

0.03

0.02

0.01

0.00

0.7

,/,,,,, ,/

\\ o.~

\ 0.60.0/8

,~, ,~_~~

0.2 0.4 0.6 0.8 1

Fig. 6.2a Subsonic inlet additive drag (-7 = 1.4).

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Fg.

=0.5

rhoa o

0.2 0.4 0.6

Fig. 6.2b Static inlet additive drag ('7 = 1.4)

0.8 1

194

SIZING THE ENGINE: INSTALLED PERFORMANCE 195

special doors found on most cowls and nacelles, which open only during takeoff

to allow for both the effective reduction of M1 and better prevention of boundary

layer separation. High bypass turbofan engines are also throttled up in a systematic

manner to limit losses.

The sharp leading edges of supersonic inlets, such as those of the AAF, prevent

the suction and additive drag recovery from occurring except when M0 is quite

near M1, in which case

(hi, let

is quite near zero anyway. Equation (6.6a) therefore

can be used to provide a useful preliminary (and conservative) estimate of

dpi,tet

for

the example aircraft. The information necessary to apply Eq. (6.6a) comes directly

from the flight condition (i.e., M0 and To) and the engine power setting (i.e., A0

and

Fgcffno )

once the design point value of Al has been fixed. The subsonic inlet

additive drag of Eq. (6.6a) is included in the AEDsys Mission Analysis, Constraint,

and Contour Plot computations.

6.2.2 Supersonic Inlet Drag

Internal inlet drag.

Because signals of the presence of the inlet cannot prop-

agate upstream to warn the oncoming flow and because the flow near the inlet

is dominated by oblique and normal shock waves as well as boundary layer sep-

aration, supersonic inlets must be analyzed differently from subsonic inlets. For

the same reasons, there are many generic types of supersonic inlet configurations,

including fixed and variable geometry versions of internal, external, and mixed

compression inlets, 5 as illustrated in Fig. 6.3. As a result of this variety, there are a

proportionately large number of methods for predicting the upper and lower limits

and the actual values of

(~inlet

for each condition.

The method employed in this textbook is based upon Fig. 6.4, which corresponds

to the inlet "swallowing" its projected image (i.e., A1 > A0) and, therefore, in-

volves no additive drag in the strictest sense on the stream tube. Swallowing the

flow at each flight condition will require a variable geometry inlet, which positions

the shock system properly. We will assume that

AI isfixed at its largest required

value.

Thus this inlet will have more airflow than the engine requires at all other

~ " ' "~ Normal

xl~t I I

,/ ,' ,, ~ Normal Shock ~." / ~ Shock

..... ..._. ..... "__' .......

Internal Compression

External Compression

. ."./, ,~, ~ Normal

~,, , ,

, Shock

Mixed Compression

Fig. 6.3 Types of supersonic inlets.

196 AIRCRAFT ENGINE DESIGN

I t

Normal Shock. Bypass ~

• ." ~_I~ "~ ," Diffuser ~

Fig. 6.4 Supersonic inlet model.

flight conditions. The excess inlet airflow must then be dumped overboard up-

stream of the engine face, but at a lower velocity than it had in the freestream. The

loss of momentum of this "bypass" air results in a drag on the inlet.

Before proceeding with the analysis, a few related points will be considered.

First, the information presented in Ref. 4 makes it clear that this is not usually the

minimum drag condition. By moving the shock system forward so that some air

"spills" outside the inlet lip, a favorable trade between spillage drag and bypass drag

can be made, which results in a lower total inlet drag. The calculations done here

will therefore be slightly conservative, but they avoid much greater complexity.

Second, even when no bypass air is present, the inlet is usually designed to accept

slightly more airflow than the engine requires in order to prevent internal boundary

layer separation. The extra air flow that is "bled off" the internal surfaces and

its loss of momentum are also chargeable to the inlet system as drag. The inlet

area (A1) must be initially sized to provide the necessary boundary layer bleed

air at its design condition. Third, it will be assumed that the bypass and bleed

flows are returned to the freestream with velocities that are sonic and aligned with

the freestream velocity. This greatly simplifies the analysis and rests upon the

observation that, even after passing through a normal shock, supersonic airflows

recover sufficient total pressure to isentropically reaccelerate beyond sonic velocity

if returned to their original static pressure. This can be demonstrated very quickly

with the help of the isentropic compressible flow and normal shock portions of the

GASTAB computer program. Finally, this approach does not account for internal

friction, the disturbed external pressure field on the (essentially straight) cowl, or

the shock overpressure available for lift. With this as background, the supersonic

inlet drag is estimated to equal the momentum change of the bypass and bleed

flows or

(Pinlet

poVo(A1 - Ao)(Vo - Ve) = (A1/Ao -

1)(1 -

Ve/Vo)

(6.7)

Fgc Fgc/(rho Vo)

where Ve is the velocity with which the bypass and bleed flows leave the inlet.

Since Tte=

Tto,

it follows that

aeMe Me TFT~ / Me[I+(v-1/2)M~I'/2

- -- - =

Vo aoMo

M00 //1 + (×

1/2)M~

SIZING THE ENGINE: INSTALLED PERFORMANCE 197

or, because Me=l

so that

Vo - Mo + × +

Y -1M2"~ U2} /{Fgc/(rhoao)}

(6.8)

(binlet'--- (~0 --1)[Mo-- (~-~t- y_.[_ 1 oJ

For typical values [e.g.,

A1/Ao

= 1.2, M0 = 1.5, and

Fgc/(moao)

= 4],

Eq. (6.8) yields

q~inm ~

0.02, which, again, is not large. Note also that

qSi,,let

approaches zero both when M0 approaches unity and when Ao approaches A l

(the sizing condition), so that

cbi,,tet

can matter only when Mo is far from both

conditions.

Once A1 has been selected, this equation can be directly evaluated at any given

flight condition (i.e., a0 and M0 ) and engine power setting (i.e., A0 and

Fgc/fno ).

The supersonic internal inlet drag of Eq. (6.8) is included in the AEDsys Mission

Analysis, Constraint, and Contour Plot computations.

External inlet drag.

You may have noticed that the external surfaces of the

nacelles of supersonic engines are often closely aligned with the direction of

the freestream flow. This is because there is an additional drag due to the pres-

sures exerted on any external surfaces exposed to the oncoming flow that would

contribute to the inlet drag of Eq. (E. 15). The most obvious example of this phe-

nomenon is the overpressure caused by oblique shock waves on the forward facing

portion of the inlet that is inclined to the flow, as in the simple case illustrated in

Fig. 6.5.

oblique shock

rh o

// fl Pe

/ . S

[ \ wedge

Fig. 6.5 Flow in the region of a wedge shaped two-dimensional inlet.

198 AIRCRAFT ENGINE DESIGN

3.0

I''''I''''I''''I'''

2.5

2.0

1 1.5

1.0

0.5

M 0 = 1.5

0.0

, , , I .... i , , i i I r , , , I

....

0 5 10 15 20 25

Wedge angle 8 (o)

Fig. 6.6 Overpressure on the external surface of the two-dimensional inlet of Fig. 6.5

as a function of 8 and M0, from the GASTAB computer program.

The contribution of the axial projection of the external portion of the inlet to the

drag is given by the expression

(ee - Po)Ae

~)inlet --

(6.9)

which, by means of the usual substitutions, becomes

(ee/Po - 1)(Ae/Ao)

~)inlet -~"

(6.10)

( F gc/thoao)(F Mo)

The term

(Pe/Po -

I) appearing in Eq. (6.10) is a function of the wedge angle

and the freestream Mach number that can be computed using the two-dimensional

oblique shock portion of the GASTAB computer program. The results of such

computations for representative values of 8 and M0 are shown in Fig. 6.6. It can be

seen that

(Pe/P0

-- 1) is insensitive to M0 but increases rapidly with 3, encouraging

designers to avoid large wedge angles.

For the same typical values used in the preceding examples, and for S = 10 deg

and Ae/Ao

= 0.2,

fbinlet -~

0.02, which is not large but worth avoiding. The inter-

ested reader can easily repeat these computations for the analogous case of axisym-

metric inlets using the conical oblique shock portion of the GASTAB computer

program. Would you expect the drag to increase or decrease for the axisymmetric

case?

SIZING THE ENGINE: INSTALLED PERFORMANCE 199

A similar analysis could be carried out for the drag caused by the oblique

expansion waves that reduce the pressure on rearward facing portions of the nacelle.

In this case, however, the absolute magnitude of (Pe/Po - 1) can never exceed one

because Pe must exceed zero. The supersonic external drag of Eq. (6.10) is not

included in the AEDsys Mission Analysis, Constraint Analysis, or Contour Plot

computations because the shape of the cowl is not generally known in sufficient

detail at this stage of design.

6.2.3 Exhaust Nozzle Drag

As in the case of inlets, it is assumed that the exhaust nozzle external viscous

forces are accounted for in the airplane drag. Referring to Fig. 6.7, which contains

a drawing of the type of nozzle to be used as a basis for drag estimation, it can

be seen that there is always a region of separated flow surrounding the point s,

which would otherwise be a stagnation point for the external flow. The external

boundary layer separation causes a pressure or "base" drag to occur, and this

constitutes the preponderance of the nozzle drag. The nozzle drag situation is

completely analogous to the inlet, in that the separation prevents attainment of the

high pressures near s that would compensate for the suction on the remainder of

the surface and result in zero pressure drag.

An interesting and useful correlation method presented in Ref. 7 has been de-

veloped for predicting the exhaust nozzle pressure drag

Dnozde "-- (P -- Po) dA >_ 0 (E. 16)

in subsonic and supersonic flows, based on the "integral mean slope" (IMS) as

defined by the equation

fl A9/AIO d(A/Alo) d( A

IMS -- (1 - A9/Alo) d[x/(Rlo - R9)] \~10.] (6.11)

I ~x

Fig. 6.7 Axisymmetric exhaust nozzle model.

200 AIRCRAFT ENGINE DESIGN

0.20

• Symbol

[]

0.16

0.12

0.08

0.04

Fig. 6.8

corner

Mo=0.7

0.5

0.65 Pte/P 0 = 2.0

0.8

, _

0.2 0.4 0.6 0.8 1.0 1.2 1.4

IMS

Convergent nozzle boattail pressure drag coefficients. 4'7

0 1.6

where the geometrical dimensions are the same as those of Fig. 6.7. This equation

by itself underscores the complex nature of engine/airframe integration because

the location of the "end" of the airframe and the "beginning" of the nozzle (i.e., A 10)

is a judgmental decision.

Once the IMS is computed from the nozzle geometry, the corresponding drag

coefficient can be obtained from Fig. 6.8 when 0 < M0 < 0.8 and from Fig. 6.9

plus the empirical relationship

Co(Mo) _ 1-1.4exp(-M 2) (6.12)

Co(1.2) ~o 2- 1

when 1.2 < M0 < 2.2. When 0.8 < Mo < 1.2, the drag coefficient is given in

Fig. 6.10 as a function of Mo and L/Dlo.

Lacking actual nozzle contours (i.e., D vs x), it is impossible to precisely evaluate

the IMS using Eq. (6.11). Real progress can be made, nevertheless, by evaluating

the IMS for the general family of nozzle contours described by

D -1-(1-D9)(x) n

Dto DI0 L

Z {, (1 O9)xn}2

A,0 = (9

(6.13)

The shape of these "boattails" for D9/Dlo = ~ and a range of n is shown in

Fig. 6.11. Note that for all n > 1 they have a smooth transition to the fuselage and

avoid sharp corners, which often cause large drag. Because of the need to store

supporting structure, actuators, and coolant passages, as well as to avoid sharp,

hard-to-cool trailing edges, practical nozzles have 1 < n < 3.

SIZING THE ENGINE: INSTALLED PERFORMANCE 201

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

1. ZERO ANGLE OF ATTACK

2. INVISCID FLOW

Ag/AIo=0.75

CONICAL AFTERBODIES OF / . 0.50

CIRCULAR SECTION ~,~/~ 0.25

////

CIRCULAR ARC AFTERBODIES /1~

OF CIRCULAR SECTION /

A~A,o=O9O ~ /4 ~°°°

/.7

012 014 o'.6 018

EXIT TO MAXIMUM

AREA RATIO

¢n 0.70

• 0.50

• 0.26

1.0 1.2

IMS

Fig. 6.9 Comparison of IMS correlation and theoretical wave drag for isolated

axisymmetric afterbodies--Mach

1.2. 4'7

Substituting Eq. (6.13) in Eq. (6.11) and integrating leads to the result that

IMS

(Dlo -

D9/L)(1 - D9/DIo)

{ (1-D9/Dl°)2}

4n 2 1 2(1 -

Dg/Dlo) +

--(I+D9/Dlo)

2n -1 3n -1 -4n Z1

(6.14)

The behavior of Eq. (6.14) with n and

D9/DIo

is shown on Fig. 6.12. The

inescapable conclusion is that one must hardly be concemed with the exact shape

of the nozzle at this point in the design and that

This result is intuitively appealing because the adverse effects of separation should

increase both with the slope of the external nozzle contour and the amount of area

change.

This enormously simplifying step makes it possible to estimate the

IMS

from gross nozzle design parameters, obtain the drag coefficient from either

0.2(

0.18

0.16

0.14

0.12

CD P 0.10

0.08

0.06

0.04

0.02

~:NASA~D-7192 ~r-~± ~ !l /~

d/D = 0.5 HI

.l:, c.

/~,9/P 0

= 2.0 . I N::I'O"',O

L/D = 0.8

11770-'-~~J

I I I I I I

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Fig, 6,10 Experimental pressure drag coefficients of some circular--arc boattails, 7

1.0

0.9

0.8

D / Dto

0.7

0.6

0.5 t_

0.0

~ ' I ' I I ' l ' I

0.2 0.4 0.6 0.8 1.0

x/L

Fig, 6.11 External nozzle contours,

202