v

AI AA/SAE/AS M E/ASE E

June 24-26, 1991 / Sacramento, CA ~ 27th Joint Propulsion Conference

AIAA-91-2007

Wake Ingestion Propulsion Benefi

L. Smith, Jr. GE Aircraft Engines Cincinnati, OH

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 370 L'Enfant Promenade, S.W.. Washington, D.C. 20024

WAKE INGESTION PROPULSION BENEFIT

Leroy H. Smith, Jr.* GE Aircraft Engines

Cincinnati, Ohio

Abstract I t is wcll known that the efficiency of propulsion is im- proved if part or all of the propulsive fluid comes from the wakeofthc craft being propelled. In thispaper this propul- sion hcnclit is quantified in terms of wake parameters and propulsor properties. The formulations apply directly to unducted fans or propellers, but the conclusions are also relevant to ducted propulsors.

It is found that the power saving is greatest when the propiilsor disk loading is high, whcn the wake form factor is high (flow near separation), and when the propulsor de- sign is such that the wake profile k n d s to be flattened as it passes through the propulsor (high wake recovery). Exam- ples are given showing that thc benefit can be in the 20% range in some cases. Propeller design parameters that lead to high wake recovery are also given.

" 1.0 Introduction It has long been known in the field of marine propulsion that the propulsive efficiency is improved whcn fluid from the wake of th r craft is used as part or all of the propulsive stream. Betz' explains this and points out that , with wake ingestion, the power expended can actually he less than the product of the forward speed and craft drag. Wislicenus and Smith', Wislirenus3, Gearhart and Henderson", and Bruce e! d5 conducted design studies of propulsors em- ploying wake ingestion aimed mainly a t the propulsion of submerged bodies, and wake-adapted propiilsors are com- monplace for torpedo and other marine applications.

For aircraft propulsion, wakc ingestion appears some- what less beneficial. With wings the wake is spread out, so it is harder to capture a substantial portion of it with thc piopnlsor. Another drawback i s the reduced density and total pressure of the air in the wake; this is mainly a disadvantage to the core engine which is de-supercharged and therefore has to be made larger. However, there are aircraft applications where wake ingcstion is clearly bene- ficial. One of these is the cruise missile where the concen- tric aft-located single-engine propulsor can capture most of the hody wake, and rl hottom inlet can supply the core engine with largely unspoiled air. Unpublished studies of ,inducted fan propdsors on cruise missiles have shown at least a 7% power reduction due to wake ingestion. The au- thor thinks it is likely that other worthwhile applications can be f o n d .

~ . ' Manager, 'rurhomachinery Aerodynamics Technology

'~--/ Fellow ASME

Another category in which wake ingestion is of interest i s in the treatment of the drag of appendages such as sup- port struts or control surfaces. If the appendage wake fluid passes through the propulsor, part of the drag i s offsct by an improvement in propulsive efficiency, with a likely hen- efit for the system as a whole.

The wake ingestion propulsion benefit will hc quantified in two ways. First, a power saving coefficient will be de- fined. This is most relevant to the appendagc category of cases. Then a propulsive efficiency with wake inges- tion will he derived; this will be of greater interest when a substantial portion of the propulsor thrust comes from the propelled craft's wake fluid

Nomenclature A c

CT Crit D

H 12 K m PSC P P p, PT R

T 5

U V v3 2/ 6 6' A 7 7)KE

= propulsor disk arm, Figs. 3 and 17 = blade chord = local thrust coefficient, (T /A) / ;p[P = thrust-loading coefficient, TJ$pV:A = viscous drag of the part of the craft whose wake

is to be ingested = wake form factor, Eq. (14) = wake pseudoenergy area, Eq.(ll) = wake pseudoenergy factor, Eq.(15) = mass flow rate through the propulsor = power saving coefficient, Eq.(23) = static pressure = shaft power = propulsive power, Eqs.(4) and (21) = total (stagnation) pressure = wake recovery, Fig. 3, Eq.(16) = blade cimimfcrentid spicing = t d a l propulsive thrust of the propulsor system.

This includcs any thrust or drag pressure forces on the craft that are induced by thc action of the propolsor.

= local hlade speed = velocity relative to the craft being propelled = axial vrlocity in the je t ; axial means thc

free-stream flow direction .= distance across wake = wake arca, Fig. 3 = wake displacemcnt area, Eq.(Q) = wake velocity defect, Fig. 3 = propulsor efficiency, V, T I P = axial kinetic energy efficiency, Eq.(7) = propdsive efficiency, Eq(5) and footnotr = wakc iiiornentuni area, Eq.(lO)

1 Helease A: "Copyright c 1991 by the American l n s t i i u t c

of Aeronautics and Astronautics, Tnc. A l l r i g h t s reservcd."

cascade deviation coefficient, Eq.(58) fluid density cascade solidity, CIS flow an& measured from circumferential

Y

direction, Fig. 12 cascade zero-lift direction, Fig. 12 local advance coefficient, V,/U streamfunction, Fig. 17

becomes for this case, using Eqs.(2) and (4 ) , the wrll known Froude efficiency

W In applying this model to an actual rase at a givcn t,lrrust

i t is necessary t o recognize that the actual shaft power is greater than the propulsive power for several reasons:

11) The flow throueh the ~ r o ~ u l s o r ivill incur V ~ S C O I I S ~, ~ . I

losses and, perhaps, shock losses. (2) Some of the kinetic energy in the jet may r i o t ijc

axial. This is certainly the rase if the propulsor is a siirglr~ rotation propeller.

(3) With a finite number of bladcs a propeller will have an induced loss. We incorporate all of these reasons into another efficiency relating the propulsive power, Pp> t o the nctoal shaft 1)owcr,

Snbsc.ripts

j = in far downstream je t , in Iree~streanr direction o = fnr upstream; in free-stream, ambient p = at the propulsor disk, except when defined above t = at propulsor tip ZL = in circumferential direction w = in wake D W P :: wake propulsion ideal case 1 = immediately ahead of propulsor disk 2 = iinmediately behind propnlsor disk

Superscript -____ = wake not ingested by propulsor

2.0 Analysis -~ 2.1 I'ropulsor W i t h e w a k e Ingestion

The basic model used for analvsis is the simole classical actuator disk shown in Fig.. For this model the density

Disk Area,

Figure 1: Actuator Disk Propulsor

is uniform, the flow is hxisymmetric, the static pressure far removed from the disk is uniform at the ambient value, there arc no viscous forces or mixing at the edgea of the je t , and properties across the jet are uniform. With this model it can be shown (see, for example, McCormick6) that the axial vclocity at the propulsor disk is the mean of the fur upstream and far downstream values

The thrust is the mass flow times thc increase in axial velocity

T = 7 i L ( K - V o ) (2) = p n v , (v, - v, ) ( 3 )

The propulsivc power is defined as the mass flow times the inc.rease in axial kinetic energy of the fluid

The propulsive efficiency'

(4)

and the overall propulsor efficiency then is

(7)

In this paper the focus will be on propulsive efficiency effects. It is thereforc tacitly assumed that T K E is more or less unchanged when a wake is ingested into a givm type of propulsor. This assumption is most questionnblc for single-rotation propellers where additional exit swirl ki- netic cnergy will likely he imparted to the wake fluid. A rational treatment of ~ K E changes with wake ingestion for single-rotation propellers could be undertaken, but that is beyond the scope of this paper.

2.2 Wake Properties Fig.2 - shows a body and i ts visc.ous wake. For the pnrposr

d

V"

Pw = Po

Figure 2: Body and Wake

of identifying wake properties we specify that the static pressure across the wake is constant at the ambient vnluc. This removes potential field effects so that the wake will be directly related to viscous drag. This will hc disrimrd further in Section 5.

The following wake integral properties will be e n i p l i ~ ~ ~ d in the analysis:

Wakc displacement area, 6' = l6 (1 .~~ 2) dA (9)

* The author has been unable to locate i( universally accepted definition for the term "propulsive efficiency ". The definition adopted here i s similar to that used by Wislicenns3 and Gearhart and Henderson4 except losses caused by caring external surfaccs and downstream swirl have not been included. Since with wakr ingestion the propulsive efficiency can be greater than unity it is soindimes called "propulsive coefficient ". In the definition of propulsirr r f f i ~ ciency uscd by naval architects the numerator contains the total craft rc&tance (which equals T herein), and t h e denominator i s thc full shaft power: thus it matcher the propulsor efficiency defiard licrciii by W 8 ) . V'

2

The propulsor thrust is the sum of the thrusts produced @ = J i z : ( * -- 2) d A (lo) by the non-wake and the wake fluid streams: Wake momentam area,

W

Wake pseudoenergy area, k = s" (1 - 2) dA(11)

whcre dA is an area. elcment at the plane where the in- tegrals are being evaluated. Analysis of a control volume mntaining thc hody gives us the body drag

v, + v, 6 T = P ( A - J ~ ) ~ ( T + K ) + ~ J v ( ~ ~ - ~ K ) ~ A (19)

where d A is an area increment upstream where the in- tegral is evaluated. To evaluate Eq.(19), is ohthined from Eq.(18) and 4, is obtained from Eq.(16). Using also Eqs.(S),(lO), and (13), Eq. (19) can he simplified to D = p J 6 V, (V, - V,) d A (12)

Using Eq.(lO) this is

D = pV:B (13)

Wake shape parameters that will he used are

S' Form factor, H = - (14) a

(15) k

Pseudoenergy factor, K = - a 2.3 Propulsor With Wake Ingestion a.2 5hows a wake passing through the actuator disk

w,,kr ~ t , . , . ~ ~ ~ . ~ ~ . I< .~ L ~ 'J %

Figure 3: Propulsor Ingesting Wake W

propulsor. For the analysis the wake fluid maintains its identity, and viscous shear stresses are neglected. The flat- tening of the wake is measured by the wake recovery, R, as defined in Fig. 3. The wake is flattened ( R is greater than zcro) for two reasons: (1) a propulsor has a natural tcadency to add more energy to fuid that approaches it with lower axial velocity and (2) even if the wakc fluid got thc same energy from the propulsor as did the rest of the propulsor stream, its axial vclocity defect would be reduced so as to keep thc same kinetic energy defect. The actual value of R depends on many propulsor parameters, some of which will hc evaluated later. For now, w e keep R in the analysis as an independent parameter. Also, for simplicity wc as~ume that R has thc same valuc for all wake stream- lines, not jus t the wake center strmmline shown in Fig. 3 . Thus,

vj - v,, R = l - - v, - v, is a constant for all wake streamlines.

I t will be convenient to employ a wake displacement thickness at the propulsor face even though the static pres- sure is not uniform there. With constant density the wake fluid volume flow is the same at the propiilsor face as it is far upstream. Therefore

(16)

&(So .~ 6;) = V0(6 - s') (17)

In a similar fashion the propulsor propulsive powcr,

is evaluated using also Eqs.(11),(15), and (ZO), and after considerahle algebra we obtain

2.4 Powcr Saving Coefficient Consider that the body shedding t h r wake to bc ingested

has x certain draa. D: an eaual thrust must be urnvided to -. , propel it. In one case its wake goes through the propulsor, and in another case it does not. We pos tuht r that the propulsor disk area is thc same in both cases. Anticipating that the propulsive power will he less when thr wake is ingested, we define a power saving coefficient

where the prime denotes the ease when the wake is not ingested. Thc denominator of Eq.(23) is seen to he the propulsive power required to propel thc hody when i,he wake is not ingested.

The power saving coefficient is defined in terms of prop,& sive powers. If we assume that ~ X E is the same with and without wakeingestion, then it can also he applied to shaft powers; this can he seen by dividing bath the numerator and denominator of Rq.(23) hy ~ K E .

The total propulsor thrust, T , madc up of D plus whxt- ever other thrust is needed, is the same for both cases:

T' = T (24)

For thr case when the wake is not ingested, Eqs.(20) and (22) simplify to

(26) 1 P; = 5T'(y + v,)

Substituting Eqs.(26) and (22) into ( 2 3 ) , using Eq. (24) and also using Eq.(G) with primes added, the power saving coefficicnt becomcs

Employing Eq.(l) this hecomes V! - V. T V,(2 ~- R) v, psc = _1__ - [- -~ 1 i~ R ( l - K ) ] l$ '+V,D+ y+V, V,

(18) (27) vj '- '/"(6, -. 6;) = V,(S - 6')

2

3

Further manipulations are needed to express v: in terms of V, and other convenient variables. Substituting Eqs.(25) and (20) into (24) and employing Eq.(13) we obtain

B 6; 0 = y2 + 2Kz-R - (v," - V:)-- A 0 . 4 (28)

Manipulation of Eq.(20) using (13) leads to

In order to determine 6; we study the deformation of the wake as it reaches the disk. The actuator disk result al- ready given by Eq.( l ) can be applied to the individual wake streamlinrs

(30) V,, + v, v,, = 2

T h m , using Eq.(16), we find

By definition,

Using Eq.(31)

for sliallow wakrs. It is necessary to assume shallow wakes so that d A j d A , will be nearly constant for all streamtubes; otherwise Eq.(34) is not strictly true. Using Eqs.(34) and (9), Eq.(33) then yields

For small d u e s of D I T , V/ - V, is nearly zcro and the first term in Eq.(27) becomes indeterminent. This case is treated in the Appendix.

will be used to represent disk loading For presentation of results, a thrnst-loading coefficient ~\

LJ

(381 T

ZP 0

CTA =

Using Eqs.(24) and (25) this is also

2.5 Propulsive Efficiency With Wake Ingestion The concmt of DroDulsive efficiency is the same whr th r r . . .

or not there is wake ingestion; thcrefore Eq.(5) still applie,. Using it with Eq.(22) yields

As explained in Section 2.1, in order to obtain the overall propulsor efficiency, which is based on shaft power, i t i b

necessary to multiply ?lp by ~ K E , which is thc efficiency f%r converting shaft power into jet axial kinetic energy flux.

2.6 Special Case: Wake Propulsion Ideal For a case when all of a craft's drag is viscous drag ,

the highest propulsive efficiency will be attained whcn o n l y wake fluid passes through the propulsor and when cacti streamline has its axial velocity brought back up to tlic free stream value. We call this the wake propulsion idral C ~ S C , and for it w e have

& - A , R = l , V , = V ' - V , , T - D ( 4 1 )

Using Eqs.(41), (27) becomes i/'

and using Eq.(39) this becomes

With 6, = A , T = pl/,zO, and B = P/H, Eq.(38) becomcs

(44 ) 2 6 '

CThlWP= -- H 6,

Further manipulations using Eqs.(l),(l4) and (17) lead to

2- 6' - YH(Z.-R)V:[ 1 - $ ] (36) Using also Eqs.(lS) and (36) , (44) can be manipulated to B (V, .!. Va)2 1 - 036'

V,+v, 6 s' I? s'\

With Eqs.(27), (2S), (29), and (36) we have the power saving coefficient expressed in terms of

R

V,/v, A measure of the blading capability to attenuate wakes A measurc of the disk loading A measure of the wake quant.ity being considered Mensurcs of the wake shapr

DIT

H, K,S'JS

( 4 5 ) 6 \" 6 1

H ( 1 - % ) CTh 1 1VP =

This interesting result shows that, for the wake p ropu l~ i o n ideal c.ase, the thrust-loading coefficient depends oiily on wake shape parameters. Furthermore, substitution of (45) into (43) gives an expression for the power saving c o ~ efficient that depends only on wake shape parnmetrrs.

Using Eqs.(41), (40) becomes

a very simple expression for the propulsivc efficiency for the wake propulsion ideal case.

4

v ~~~~~. . ~

work was done on this in the 1950's a t NACA, and G; from Lirhlein and Roudebush7, gives a correlation for two-

"..kt YO,," Fact, , , , ),

Figure 4: Pseodocnergy Factor Data For Airfoil Wakes, Ref. 7

dimensional airfoil wakes. Also shown on Fig. 4 are results for several analytical wake representations, and it could be argued that any of them fits the data adequately. For our purposrs we adopt the error curve shape, Fig.. It's arialytic.al represent,ation is (Liehlein k Roudehrrsh7)

where

b, = In [ l o 0 4 (1 - y?)] For this shape the pseudoenergy factor is

where 2

a = - &

and the displaccmcnt thickness ratio is

where b = In [loo&? ( T)] H - 1

Eqs.(48) and (49) will be used for simplicity when the numerical results presented in Section 3 are generated, even though such a tight correlation as shown in Fig. 4 would probably not he found for three-dimensional body wakes.

""4

D l s l a . c c firm WlkC CcnLvrlii lc

Figure 5: Error Curve Wake Profiic

3.0 Numerical Results 3.1 Power Saving Cocfficient Evaluation

The calculation procedure is as follows 'l'hrrr of the independent variables, R! D I T , and H are given, and rev- era1 values of %/V, are also specified. Eqs.(48), (49): (%), (29), (28), and (27) arc then applied to obtain pompr saving coefficient values. The corresponding thrust-loading roef- ficients arc calculatcd from Eq.(39). Eq.(3i) is evaluated, and cases for which AJ6, < 1 are climinatcd bez i~me they violate the assumptions of the niodel shown in Fig. 3 .

was constractcd for cases where the Body drag is murhless than the prupu l so~ thrust. A casr in point i s the

' , h r t ~ s t . ~ , o 3 d i c v C n e f f i i L i n t , C.i,,

Figure 6: Effects Of Disk Loading And Wake Form Factor On Power Required To Propel That, Part Of The Cmft Whose Wake Is Being Ingestrd

support strut for an unducted fan engine aft-mounted on an aircraft fuselage. With this case in mind, the abscissa scale has been double labeled for typical GE36 operating points. It appears that perhaps 15.20% of the strut drag would be recovered a t high Mach operation depending on the wake form factor.

The dependency of PSC on H has physical significancc. As viscous wakes move downstream, shear stresses tend to flatten them. This reduces their form factor and less power will be saved. The message here is that , to ma.xi- mise efficiency, the propulsor should be positioned to ingcsf B viscous wake before the wake has had m u c h time to rlis- sipate. The same phenomenon is thought to prtiall:. ex- plain the experimental observation that niultistage t u s k - machines have improved performance whcn thr axial shps between hlade rows arc reduced.

Fig. 6 also provides help in estimating the improved in- stalled performance of a wing-mounted pusher propeller that acts on the wake fluid of part of the wing compared to that of a tractor propeller whose wash increases the drag of the wing.

The ability of a propulsor to add more energy to the low velocity parts of a wake and thus to recover the flow to a more uniform state is measured by the propulsor's wake recovery, R, dcfined in Fig. 3. In R has been varied

',~~~,,..,~,.":,~~,,~ C C W ? ~ * C , ~ " , , u.l,,

Figure 7: Effect Of Wake Recovery

over a wide rangc. As expected, the power saving is great^ est whcn It = 1, the case for which the jct is uniform. It is remarkable, Irowever, t,hat the recovery can differ signif icantly froni unity without the power saving being much affected. This is fortunate because an accurate prediction of R for a given propulsor is difficult. Approximate meth- ods for estimating R for propeller-type propulsors are given in Scction 4.

A s dcfined, the p w c r saving cocfficient is normalized by thc power requircd to propel the body (or thc part of thc body) whose wake is to be ingested. In the two pre- ceeding fignres that powcr was small compared to that of the wholc propulsor, since DIT = 0.03. In w e con-

lhrurf- l .oading c . ie f f ic ies l , c.,,,

Figure 8 : Effcrt Of Quantity Of Wake Being Ingested

sider also cases for which a large part of thc total craft wakeis ingested. The curves are terminated at the thrust- loading coefficients for which the disk areas have hecomc small enough that they equal the wake areas. Whcn P / T

whose thrust just equals the craft's visc.ous drag; lor this case PSC is a fraction of roughly all of the power so i ts nu- merical values are lower. For such cases it is usually more useful to employ the propulsive efficiency with wakr inges- tion, rather than the power saving coefficient, t u estimate power savings. This is done in the following subsection.

= 1, all of the craft's wake passes through the propulsor, \

V

3.2 Propulsive Efficiency Evaluation We proceed as in the previous subsection, sprrifying R,

DIT , and H values and a range of V,/V,. Eq.(40) is used to calculate T ~ , and Eq.(39) is used for CTA, liiniterl by Eq.(37).

For the examples shown in Fig., H is set at 1 .R, ivliicli

'Tl>rL>sr-L"<zdi,'* CDErii

Figurr 9: Propulsivc Efficiency Gains From Wakv lrigcstion

may be representative for craft such as cruise niissilcs. R is 0.8, which is probably attainilble for most propulsors dcsigned for wakr adaptation except single-rotntion pro- pellrrs. But to show that the efficiency is not very srnsitive to R, R has also been given other values for 1117' = I 0.4. The effcct of R is somewhat greatrr for larger l1,"r v;doc-s.

The DIT = 0 curve in Fig. 9, whirl1 applics w l i c n no wake is ingested, is the familiar Fro& efficiency variat,ion showing a significant loss in efficiency as thc propdsor is made smaller. But when a large part of thc crnft,'s wakc is ingested by the propulsor, there is niiic.h lcss iiicciitive to keep the propulsor large. The message hrrc is that , for best efficiency, the propulsor should be positionrd and sized to ingest i ~ s much wake fluid as possiblr ( i n ~ w a s r D I T ) , but after that , making it still largcr docs no t pay off in propulsive cfficicncy and would have otlicr ndvcrsr effects such as increased weight.

d'

3.3 Wake Propulsion Ideal Case Evalnat,ion F h r analysis prcsented in Section 2.6 for this caw, iulirrr

the propulsor just restores the momentum dcfcd in thr wake to produrc a thrust just equal to thc drag, indiratcd that thc thrust-loading coefficient of suclr mi idml prop111- S O T dcpends only on wake shape parameters. This rclat,iori- ship, Eq.(45), is shown in Fig. lo(&). The corrrsponding ideal power saving coefficient, Eq.(43), and idcal pmpnlcivc efficiency, Eq.(46), are shown in Fig. lO(b). Tlir potcntisl power saving is ern to be quite significant. Fig. I O again emphasizes the benefit of placing thc propulsor forward i n the wake where the form fador is highest. Tlrc Iwnrfit is not only in efficiency, but also, as Fig. lO(a) indii.ittrs, in

_ _ ~ _ _ _ ~ ~ ~ _

__-

red,lrrd propulsor size. ~~/

U

wok. rrrm Ylrcer. / /

Figure 10: Wake Propulsion Idral Case

4.0 Wake Recovery Evaluation For a wake-adapted propulsor design, which wonld be done for a cruise missile or torpedo, the blading can he shaped to ~ S S U I C a good wake recovery (except for a single-rotation propeller where hub swirl could not be removed'). How- ever, for circumferentially varying wakes (such as the strut wake that was desc.ribed in connection with Fig. 6 ) it is not obvious what the wake recovery might be. Although it was shown that the power saving Coefficient and propulsivc effi- ciency are not very sensitive to wake recovery, some method for estimating it is needed short of detailed calculations of the flow in the propulsor blading. Some approximations are therefore develoDed in this section usina the actuator

~

disk approach. - 4.1 Element Characteristic Slope For each radial element of a propulsor the characteristic

diagram shown in Fig.ll can be drawn. Here CT is the local thrust coefficient

where U is the local blade speed and T / A is the thrust

I

Local Advance Coefficient,

Figure 11: Propeller Blade Element Characteristic

per unit area at that radius of the actuator disk. The local advance coefficient is

(51) V, $4= - u

Fig. 11 would normally be generated from tests or analyses of a free-running propeller with undistorted inflow. We can use i t , however, to estimate what would happen if locally around t,he circumference the inflow velocity were reduced

' Unless pz~t-span stator vanes ahead of or behind the propeller ~ _ _ _ - - ."/ are employed.

by dV, as in a body wake. The increase in local thrust coefficient would affect the jet velocity V;. Referring back to Eq.(16), we see that the wake recovery would then be given by

Using Eqs.(l) and (3), Eq.(50) bccomes

Differentiation of Eqs.(53) and (51) with U constant then allows (52) to be written

24+ $$ W F 3

R = l - (54)

which can be used when the element characteristics are known.

The analyses in the following three subsections can pro- vide Some guidance for cases for which the local element characteristics are not know.

4.2 Lightly Loaded Single-Rotation Propeller The restriction to light loading is a consequence of the

assumption that the &tic pressure in the jet is the ambient value. The swirl component velocity Vu is included in the analysis to obtain the wake recovery, although the wake recovery definition involves only the axial component 4.

The power added per unit mass Row is, according to t,he Euler equation of turbomachinery,

P / m = UV, (55)

This power is absorbed by the fluid and shows up as an increase in stream kinetic energy

(56) 1 2

P/m = -(by i- v: - V,2)

Equating these powers, and employing the geometry of the velocity triangles at the actuator disk shown in Fie., we can obtain

(57) 1 4

~ z + - ( V , + V , ) 2 C O t Z ' p Z ~ V ~ = U Z

This will be differentiated to obtain dT$/& for substitu-

Figure 12: Velocity Tiangles At Actuator Disk And R l a d ~ ing Cascade Parameters

tion into Eq.(52) to obtain I?. We will also employ

dcot 'pz

dcot 'pi L,=-

where I/ is a constant depending on cascade geometry. Us- ing Weinig's flat plate cascade theory ', the author has found, to good approximation,

)] (59) = cxp [-To (sin'p' + 1 - sin 'p'acscp.

1.28

where v' is the zero-lift direction and 0 is the cascade solidity as shown in Fig. 12.

Ea.(57) is now differentiated with U constant using ~q. ( i i ) and

(60) 2U

cot $0, = c_

After much dgcbra. and using also Eqs.(51)-(53), we obtain

v, + V,

This is shown graphically in w. In constructing

m

l 0 C i l ,,d".,,W,. C, , c i i l r l , . " l . 0

Figure 13: Wake Recovery For Single-Rotation Propeller Uladr Elements

Fig.13 it was assumed for simplicity that v* = pz, and since q2 can be determined from 4 and CT, R depcnds only on 4, Cr, and u. The airfoil lift coefficient can also be ralcolated from these variables, and the lines in the figure have been terminated at the locations where the lift coefficient reaches unity.

The most noteworthy feature of Fig. 13 is that the wake recoveries fall off to low values a t high advance coeificients. This means that the hub elements of single-rotation pro- pcllers will have poor recoveries because of their low blade speeds. Downstream part-span stator vanes could be used to improvc recoveries there, and they need not necessar- ily be placed around the whole circumference, but perhaps . . just in the wake region.

4.3 Iligh Solidity Counter-Rotating Propellea! F O ~ riiridicitv \ye ass-eds are eaual

I " ~ ~~ ~

and that the two rotors are very close together so that there is no chang? in properties between the two actuator disks. We proceed as in subsection 4.2 and arrive at an cquation n n a l o u p u s to Eq.(57) except that it is more c.omplex and contains the exit flow angle from the aft rotor as well as that from the forward rotor. Because of the complexity we assume that thc exit flow angles remain constant as V, is varied; thus the result applies only to rotors that have high solidity. The final expression simplifies to

This is plotted in Fig. 14. Although the downward trend with increasing a d i c o c f f i c i e n t is still apparent, the wake recowry values ace considerably highc-r than for high

L0Cll *d"mcc c , , c E * * c i ~ ~ , ~ , + Figure 14: Rotating Propeller Blade Elements

solidity single rotation QrOQelleiS a t representirt,ivc advancc

Wake Recovery For High Solidity Coruitcr

coefficientsand thrust coefficients. - 4.4 Wake Recovery For Thin Shallow Inviscid In the preceding sections the wake fluid has hecn as-

sumed to be guided by the blading in the same \my as . the free-stream fluid. A better approximation for casts i n which the wake is thin compared to the blade spacing is given in this subsection. The approach used is tlrnt de- scribed by Smith' and shown in !*. Recnusc of thr

'd

Figure 15: Thin Wake Passing Through A Rotor

circulation around the airfoils, particles travcrsc thr s n c ~ tion side Errstet than the pressure side, and a wvakr torus into disconnected segments as it passes through a rotor.

Across the rotor we assume the flow is two-dimensional, and therefore the vortic.ity in the wake, which is prrpen- dicular to the figure, remains constant as thc particles pro- ceed. Consider the fluid in a segment of thc wvakc L B upstream. Later that same fluid appears as scgmcnt C- D downstream. Since the segment length has irirrc.ascd, its width has decreased proportionately, and with constuit vorticity the wake velocity defect has decreased hrraosr it is proportional to the wake width. Formulas givrn p r r v - ously by the author" can be used to deduce thc scgiiicnt length increase. This is a piece of thc wake rerovcry, Init we need to also consider wake changes in the upstrririn and downstream regions where the flow is accelerating. I n those regions it is appropriate to assume that the flow is accrlcc- atirig because the streamsurfaces are contracting itnil tlirrc- forc the flow is not two-dimensional and thc vorticil,g is not constant. But v e can use the IIeImhola law that statcs that the vorticity is proportional to the vortex linc IcngI,h, which is proportional to the streamtube lamina thickness, which is inversely proportional to the axial velocity. A l s o assuming for shallow wakes that the tangential componmt of the wake width is constant in these regions, we can cal- culatc thc wake velocity defect changes there. Only tlic axial component of the final wake defect in the jet is risctl to calculate R. e

T h r analysis just outlined and the final expressions are presents

present. This choiceimplies that the result will be the same in either case. The result will only be the same if the po- tential field of the propulsor has not affected the upstream viscous losses. This is an important point, because often the acceleration of the flow into the propulsor will pre- vent or minimize flow separation losses on craft with blunt

rather complex and will not bc given here.

. ~~

::, 2 /1 afterhodies'. In fact, we can conceive of designing blunt e . 8 - . ~~~~ __ ~ craft that only avoid massive flow separation because their

v

propulsors ingest their boundary layers ( W i s l i c e n u ~ ~ , ~ ) . So we should use wake data obtained with the actual (or a sim-

0 1 2 3 ilar) propulsor in place if they are awilable, and recognize that if we use unpropelled data because they are all that Local *llvancc coerricient. * are available, we are tacitly assuming that the presence of the propulsor will not &ffcct the

It should be emphasized that any benefit in c r a f t Figure 16: Wake Recovery For Single-Rotation Propeller Blade Elemmts Ingesting Thin, Shallow, Inviscid Wakes

Iriscous drag.

the results obtained. I t is interesting to note that the rc- covery is now increased with increased loading. Thai is because the wake fluid is now deflected by the potential flow pressure field of the rotor blades rather than being guided by the hladr surfaces. The blade chord length does not enter the analysis, only the blade circulation and blade spac.ing. However, it is unreasonable to use this method with high solidity because then the assumption that the bladc surfaces don't guide the flow would dearly be invalid. T h e lines on the figure have arbitrarily been terminated for a lift coefficient of unity with a solidity of 0.8.

5.0 Method For Application In conducting the analyses i t was assumed that the static pressure was constant across the wake, and that the veloc- ity variation in the wake was caused by viscous rather than potential field effects. In actual applications there will be

static pressure field in the wake a t the location where the propulsor is to he placed (and a. different static pressure field there when the propulsor is in place and operating). Because of thcse pressure fields, the actuel wake velocity distributions are insufficient and inconvenient for use in our analysis, which is basically concerned with energy fluxes.

When designing a propulsor far a craft for which there will be significant wake ingestion, such as a torpedo or cruisc missile, the first step is to determine the flow field a t the location where the propulsor is to be placed either from model tcsts or from a viscous flow analysis. The most im- portant property of the flow field at the propulsor location is the distribution of total pressure, because that relates to the viscous losses that have occurred. A procedure for applying this information is given in Fig. 17. __

Q 0 ,in,iy

/ 3; i tabi i sh r o r ~ l pressure V B . sr iaamfuiwf ion a t pmpulsor p l m e w i t h (or withour) prapulsor p r ~ r m z from ncnrum~lc, ,~

nmbienr s t e r i c p'r"sYrc in v r i e .

B" ares dnq dh - d*

0 Dclcrmi.l: 4*, 0 , k from dirtribation .E@

Figure 17: Procedure For Applying Measured Or Calcu- lated Wake Data

17 that the procedure can he applied using data either with (or without) the propulsor

Note in Step 1 in Fig. ,..A

viscous h u g reduction resulting from wake ingestion is in addition to the benefit descrihed in this paper.

When a propulsor is added to a craft, it changes the pressure field on the body surface, and thus one might say it changes the craft's drag. From a propulsive effi- ciency standpoint this is neither had nor good, since the momentum and energy fluxes far downstream, and the cor- responding far upstream fluxes determined as outlined in Fig. 17, determine the propulsive thrust a d propulsive power that dctermine the propulsivc efficiency. For struc- tural design purposes it is important to know where the forces occur, but for the propulsive cfficiency determina- tion it is not. It is noted in passing that in the marine propulsion cornnionity it is commonplace to we the term "thrust deduction" to represent the drag increase of the craft associated with the flow induction field of the propul- SOI and the corresponding thrust increase needed on the propulsor to balance it.

17, the streamline that passes the tip of the propulsor disk has been indicated with the symbol t . Al- though the total pressure distribution shown in Panel l and thc velocity distribution in Panel 3 should not be much different whether or not the propulsor is in place, the tip streamfunction will be different, and so will br the tip equivalent area. The ratio of the tip equivalent area without the propulsor in place to the actual area is a mea- sure of the potential field of the basic body sincc it only differs from unity because the static prcssure is different from ambient. I t is recommended that this ratio he ap- plied to the actual area to obtain the A used in this paper.

In Fig.

6.0 Summary And Conclusions A method is presented for estimating the amount of propul- sive power that can he saved when the viscous wakc of it

body is used as part of the propulsive fluid. The prop& sive power is represented by the axial kinetic energy f lux in the downstream wake, it being assumed t,hat the wake is at ambient static pressure. The propulsive power is relatcd to the actual shaft power through the axial kinetic energy efficiency, which accounts for blading and other propulsor surface viscous and shock losscs, induced losses for opcn-tip propulsors, and wake swirl kinetic energy losses. Except for single-rotation propellers where swirl losses might change significantly, the axial kinetic energy efficiency should not he greatly affected by wake ingestion, so a percentage IC- duction in propulsive power should be nearly realized as a reduction in shaft power.

Results are presented in two formats: a power saving

+ On the other hand, it is conceivable that craft with fine nf- terbodics could have their V ~ S C O U S drag increased hy increased skin friction ahead of the propu[sor.

9

coefficient and a propulsive efficiency with wake ingestion. The power saving coefficient is perhaps more useful when assessing wake ingestion from appendanges such 8 s support struts, while the propulsive efficiency is more applicable to cases when a large part of the craft’s viscous wake is ingested. Significant power savings are found for certain cases.

Several conclusions can be drawn from the numerical re- sults:

1. The power saving with wake ingestion is great& for small propulsors, i.e. propulsors with high thrust-loading coefficients.

2. The power saving is greatest when the form factor of the wake being ingested is high. This means that the propulsor should be positioned so that it ingests the wake before the wake is flattened much by fluid shear stresses.

3. T h e flattening of the wake by reversible energy addi- tion of the propulsor is favorable. This is a property of the propulsor named wake recovery. Although high wake recov- ery is favorable, i t is not found to be of major importance. Methods for estimating the wake recoveries of propeller- type blade elements are given. I t is found that the local advance ratio (ratio of forward speed to local blade speed) is the dominant parameter. High advance ratios, such as occur near t,he huh, are unfavorable.

A procedure is given for interpreting and applying wake data from measurements or viscous analyses at the propul- SOI location on a craft. The effects of the static pressure field induced by the propulsor are discussed. It is pointed out that any benefit resulting from the reduction of bound- ary laycr f law separation losses by this pressure ficld is in addition to the benefit described in this paper.

Acknowledgements and Dedication The aut,hor would like to thank Donald M. Hill who per- formed the numerical anal,rses, Robert Henderson, Walter Gearhart, Justin Kerwin, Edward Greitzer and William Morgan who provided helpful discussions and references, and to General Electric Aircraft Engines for permission to publish this paper. The paper is dedicated to the memory of George Wislicenns, who had a lifelong interest in this subject, and who was responsible for the author’s carly interest in it.

Appendix ~. Small Wake Extreme

A5 DIT -) 0, (y’ ~ V,) i 0 and the first tcrm in Eq.(27) becomes indeterminant. But then Ea.1281 can be written. using the binomial expansion,

and rising Eq.(29)

References ‘ Betn, A., Introduct ion t o the Theory of Flow Machines , Pergunon Prcss, First English Edition, 1966, Section 59, pp. 215-217.

Wislicenus, G.F., and Smith, L.B., “Hydraulic Jet Propulsion and Incipient Cavitation”, Internal Flow Research Report 1-6, Part A, Johns Hopkins C. Mcrlr. Eng. Dept., March 21, 1952.

Wislicenus, G.F., “Hydrodynamics and Propulsion of Submerged Bodies”, J . Am. Rocket Soc., Vol. 30, 1960, pp. 1140-1148.

Gearhart, W.S., and Henderson, R.E., ‘‘Sclcction of a Propulsor for a Submersible Systcm”, J . A i ~ m n fl., Vol. 3, No. 1, Jnn.-Feb. 1966.

Bruce, E.P., Gearhart, W.S., Ross, J.R., and Trcaster, A.L., “The Design of Pumpjets for IIydrodyniLinic Propulsion”, NASA SP~304, Part 11, 1074, pp. 7 9 5 ~ 839.

McCormick, B.W., Aerodynamics of V;,7TOI, F l igh t , Academic Press, 1967 Chaptcr 4.

Lieblein, S. and Roudebush, W.H., “Lom~Spcctl Wake Characteristics of Tivo~Dimensional Cuscadc a n d I s o ~ latad Airfoil Sections”, NASA T N 3771, Octobrr 1956.

8 W ’ . e m g , F.S., Die S t r z m u n g urn die Schnwfeln, von Tuybomaschinen, J.A. Barth, Leipaig, Cemmnny, 1935.

1.

,,

Smith, L.H., “Wake Dispersion in Turhomachinrs”, J . Basic Eng‘r., A S M E T r a n s . , Vol. 8 8 , Srr . D, So. 3. Sept. 1966, pp. 688-690.

‘0 Smith, L.H., ~ e c o m ~ a r y plow in ~ x i i l ~ - ~ l o u , ~ i l s ~ m

machinery”, A S M E Trans., Vol. 77, No. 7, O c t . 1955, pp. 1065-1076.

\ J

which can be used together with Eq.(36) in the first term of Eq.(27).