w,q veq f ~

L ~ g ~ A ~ v

M I N I S T R Y OF A V I A T I O N

AERONAUTICAL RESEARCH .COUNCIL

REPORTS AND MEMORANDA

R. & M. No . 3119 (19,922)

A.R.C. Technleal Report

The Theoretical Evaluation of the Downwash behind Jet-Flapped Wings

By A. J. Ross , P h . D .

LONDON: HER MAJESTY'S STATIONERY OFFICE

I96Z

P R I C E I Z $ . 6d. NET

The Theoretical Evaluation of the Downwash behind Jet-Flapped Wings

B y A . J . R ' o s s , P h . D .

COMMUNICIATED BY THE DIRECTOR-GENERAL OF SCIENTIFIC RESEARCH (AIR)

MINISTRY OF SUPPLY

Reports and Memoranda No. 3H9" January, ±958

Summary. The dov~nwash behind jet-flapped wings of infinite and finite span has been evaluated, using the solutions given by Spence 1 and Maskell ~ for the flow in two and three dimensions respectively. The wing incidence and jet deflection are assumed to be small, and the displacement of the jet is taken into account by displacing the downwash field so that the relative distance between the jet and tailplane is correct. It is also assumed that the spanwise loading on the finite wing is elliptic, and that the trailing vorticity is generated at one particular chordwise position. The wake is considered to be a flat sheet, and so the effect of the rolling-up of the wake has been neglected.

Charts are given for the functions required in the calculation of the downwash for four specific values of jet momentum coefficient. Results have been obtained for the infinite wing and for a wing of aspect ratio 6" 0, for various tailplane positions, and varying wing incidence, jet deflection and jet momentum coefficient. A comparison is also made with some American experimental resuks 5 for a wing of aspect ratio 8.4, and the agreement is good, especially when the experimental lift coefficient is used in the calculations (since it differs considerably from the theoretical lift value, owing to the fact that the experimental wing has a blown flap).

Introduction. When considering the design of a jet-flapped aircraft from a stability and control aspect, it is necessary to have fairly accurate information concerning the downwash field behind the jet-flapped wing, particularly in those,regions where it is practicable to locate the tailplane.

The evaluation o f the downwash at the tailplane is dependent upon a knowledge of the strength

and position of the vorticity distributions which represent the wing and the jet. In his treatment

of the flow past a wing with a jet-flap, of infinite span, Spencet assumes that the incidence Of the

wing and the deflection of the .jet are small, and hence the usual assumptions of thin aerofoil theory,

in which the wing and jet are replaced by vortex sheets in the direction of the free stream, apply.

The results so obtained for the vorticity distributions on the wing and jet are used in Part I to

give the downwash at any position relative to the plane vortex sheet in the form

8e he

where e = downwash angle, r = jet deflection angle, and c~ = wing incidence. However, in the

calculation of the downwash induced at a point (P) in t h e field, it is necessary to allow for its

location relative to the actual wing and jet. To the order of accuracy consistent with the previous

assumptions, this implies calculating the downwash at a point whose ordinate relative to the plane

* R.A.E. Report Aero. 2599, received 13th March, 1958.

vortex sheets is equal to the distance of the tailplane from the jet (as shown in Figs. la and lb). The functions 0e/0~- and 0e/0~ depend upon the jet momentum coefficient Cj, and on the relative position of the tailplane; charts for these functions, and for the position of the jet, are given for various specific C a values. The downwash has been evaluated for ranges of the tailplane position, wing incidence, jet deflection and jet momentum coefficient.

For the unswept wing of finite span, with a full-span jet-flap, considered in Part II, Maskell 2 has introduced the concept of an effective wing and jet flap of infinite span, in order to obtain the strength of the bound vorticity, elliptic spanwise loading being assumed. This ~olution may be used to give the contribution to the downwash from the bound vorticity, in a similar way to that described in Part I, but it does'not account for the effect of the trailing vortices arising from the pressure gradients along the wing and jet spans. In the case of a wing without a jet-flap, it has been found 3 that the downwash is very sensitive to the relative distance between the tailplane and the wake, and that the spanwise loading has more effect on the downwash than the chordwise loading, and so the wing and its wake are replaced by a lifting line and its trailing vortices, the latter being displaced in order to keep the tailplane at the correct height above the wake. The effect of the rolling-up of the wake has also been investigated 6 for a wing without a jet-flap, and it is shown that rolling-up is not important for normal tailp'lane positions behind wings of large aspect ratio. The distance e behind the wing at which rolling-up may be assumed to be complete is given by

e/c = k'/Ct, for a wing without a jet-flap, where k' depends upon the plan-form and spanwise loading of the wing. For the jet-flapped wing, the C~ will be greater than for the normal wing, but k' may now be a function of C j, and will probably increase with increasing Cj (since the bourid vorticity on the jet will tend to resist rolling-up), so that e/c will not decrease so quickly with in-

creasing Cz~ and Cj, as might have been expected from first considerations. Thus, in order to evaluate

the contribution to the downwash behind a jet-flapped wing from the trailing vorticity, it is assumed that the majority of the load is carried on the wing, so that the trailing vortices may be considered

to arise from one chordwise position on the wing, with no rolling-up taking place. The_displacement of the jet and trailing vortices is accounted for by taking the position of the tailplane relative to the wake, and a chart is given for the downwash due to the trailing vorticity. Calculated values of the downwash are in good agreement with the few experimental results available 5, especially if the difference between the experimental and theoretical lift coefficients is taken into account. Theoretical results for the downwash on the centre-line are also given for a wing of aspect ratio 6.0, showing variation with tailplane position, wing incidence, and jet parameters.

PART I

1. Vortex Representation of the Wing and Jet-Flap of Infinite Span. The wing and jet-flap of infinite span may be represented in two dimensions by vorticity distributed on the chordal plane of the wing and the median line of the jet (assumed to be thin). The downwash relations have been solved by SpencO, using the assumptions of thin-aerofoil theory, so that the aerofoil incidence and jet deflection are considered to be small. The vorticity distributions and the position of the jet are given in Fourier-series forms, with coefficients as functions of the jet momentum coefficient Cj.

Let Uof(X ) be the vorticity distribution on the aerofoil (at incidence a to the mainstream) and 7(x) the vorticity distribution on the jet (emerging at deflection z to the extended chord-line of the aerofoil), as shown in Fig. la. The x axis is taken parallel to the main stream, and the z axis

vertically downwards, with the origin at the leading edge of the aerofoil. The chord of the aerofoil

2

is taken to be unity, so that x and z are non-dimensional. Thus the vortex representation of the flow which is in accordance with the assumptions of thin aerofoil theory is as shown in Fig. lb, with Uof(x ) located on the x axis, between 0 and 1, and y(x) also on the x axis, between 1 and co.

Then the expressions for f(x), 7(x) and zj(x), the jet displacement, as obtained from Ref. 1, are: For

0 < x < 1, x = ½(1 - cos0) (1)

f (x) = x- ~2r _ gl l°g(1 - x) + A 0 1 - cos~. + ,~=~A'~ t a n g

For

2~ Bo(1 - c°s 0) ~V-*B~(tan z'~ I + ~ / x ( 1 - .)*z~ + + ~ , 4 °-) . (2)

< X < 0% X ~ SBC 2 4) (3)

z j = r 2 ( 1 - - c o s ~ ) + A 0 log - s i n ~ - 2 s i n ~

+ ,~__1(4n 2 - 1} 2 c o s ~ s i n n ~ - 4 n s i n ~ c o s n ~

+ c¢ - - - + ( B o + ½) log - - - 2 B o s i n ~

1 + sin ~ sin 2

T (2 N

+ ,~=1 ( 4n~' 1 ) cos

U o - * l - 4 c ° s S ~ l ° g ( t a r ~ ) + 2.cosa~ 2oA,~cosnq51

(4)

+ e~ 2 c o s a ~ o B ~ C o s n ~ . (5)

2. The Downwash. The downwash induced by the vortex distributions Uof(X ) and y(x) at the point (X, Z) is given by

z) I fl v°/(*)(x - =" -(X77~)~ q- ~ dx + O~ ( X - x) 2 + Z 2 (6)

to the first order in ~ and 7 (see Fig. lb). In order to apply the results calculated for the simplified configuration (Fig. lb) to the actual

configuration (Fig. la), where the jet is displaced a distance z~(X) below the x axis, it is assumed that the downwash w(X, z) calculated for the point P'(X, z) in Fig. lb is equal to the downwash at the point P(X, z + zj) in Fig. la.

3

(78s24} a 2

A similar procedure is followed in Ref. 3, where the displacement of the wake of a finite wing

has to be considered.

In general, the tailplane will be located a distance H above the jet, as indicated in Fig. la, so

that to evaluate the downwash at the tailplane, i.e., at the point (X, zj - H) in Fig. la, we must

evaluate the downwash at the point (X, - H) in Fig. lb. The position of the tailplane is usually given as the distance along and height above the extended

chordline. If l is the distance of the aerodynamic centre of the tailplane behind the wing leading edge, measured along the extended wing chord-line, and h the height above the chord-line, when the chord is of length c, as shown in Fig. la, then the non-dimensional co-ordinates (X, Z) at which

the downwash is to be evaluated are given by

l h 32 = - + -o~, (Sa)

£ C

l h Z = - H = - ~ - - - z j (X) , (8b)

c £

where z a may be obtained from Fig. 3 *: (or equation (4)). For the numerical evaluation of the two integrals in equation (6), it is necessary to change the

variables of integration, in the first integral using equation (1) in order to avoid the infinite value

off(x) at the leading and trailing edges, and in the second integral using equation (3) to make the range of integration finite. Thus, if we write

J'(x) = f~(x) T + .f2(x) ~ "l

,/(x) ) Uo - 71(x)~- + m(x)~

(9)

then the downwash at the tailplane is given by r-

U o [ f~fl(x) ~_sin0(X - x) w = N ~ ] j o ( x - x ) ~ + N ~ ~o+

Uo ] ['•A(x) I- sin O(X - x) + ~ / J o ~ 2 - - ~ ¥ - z ~ ~0 +

L

s i n ~ ( X ~ x ) 1

s i n ~ r ( X - x ) ] (lo)

wherefi(x) s in0 andf~(x) s in0 remain finite as x and 0 tend to zero, and as x ÷ l , 0-->Tr. Equation (10) may be rewritten in the ,form

w De De Uo e = ~ - + N a ' (11)

where De~Dr and Def~ are functions of Ca, X and Z. These have been evaluated for Cj = 0.5, 1.0, 2.0 and 4.0, with 3 < X < 5 and 0 < [ Z [ < 3, the results being shown as charts in Figs.

4a to 4d.

* The suffix ~ on Cj and ~ refer to the finite-wing case considered in Part II. For infinite aspect ratio, a t - ~ and Cj e = Cj in Fig. 3 and Figs. 4a to 4d, and e~ = e in Figs. 4a to 4d.

4

Thus the procedure for the evaluation of the downwash at a given tailplane position, h/c and l/c, and given a, Cj and ~, is to calculate the functions in the following order:

(i) X from equation (8a)

(ii) z j from Fig. 3

(iii) Z from equation (Sb)

(iv) 8e/8~, 8e/Sa from Figs. 4a to 4d

(v) e from equation (11).

Interpolation will be necessary for Cj values other than 0.5, 1.0, 2 .0 and 4.0, and it seems

better to evaluate e for a range of C j, and then to interpolate the final result, rather than to

interpolate for zj, Be~ST and 8e/Sa separately.

For large X, the downwash is given by,

e(X, o) A°'r + (B° + ½)o~ C~, (see Ref. 1) X - 41rX'

so that

and

[ 8 Cz~

8~- X 4~rX

(Bo 1Tg- / 8~-- X - 4rrX

as X --> oo.

It may be noted that the value of Cz/(4rrX) for the downwash far behind the aerofoil is also

obtained when the aerofoil is without a jet-flap.

3. Results. The results for the downwash behind an infinite wing and jet-flap are shown in Figs. 7 to 11. It should be remembered that the theory is only strictly valid for small a and 7, so

that the use of the method to obtain the downwash for the larger values of ~ and ~- must wait to be justified or otherwise until experimental data are available. However, the results should indicate

the trends in the variation of downwash with the various parameters. In Figs. 7 and 8, the variation of the downwash with tailplane position is shown for two values

of jet deflection angle, ~, and two values of wing incidence, ~, for Cj = 2.0. Fig. 7 shows that on the extended chord-line, h/c = 0, the downwash decreases quite sharply with increasing distance

behind the wing, l/c, but when h = 2c, the downwash is practically constant in each case for

3c < 1 < 5c. The results have been replotted in Fig. 8 to show the downwash field (i.e., contours

of equal downwash), in the tailplane region. A comparison between the fields .for the various

and a shows that the downwash is more sensitive to tailplane position for the higher r and ~ 'values,

as might be expected. The results for the variation of e with Cj, T and ~ are given in Figs.. 9 and 10 for a representative

tailplane position, l/c = 3.5, h/c = 1.5, and also for a position on the extended chord-line,

l/c = 3.5, h/c = 0. I t will be noticed in Fig. 9a that e does not increase linearly with z for a given

Cj value (as might be implied by a glance at equation (11)) due to the correction made to the

downwash field for the displacement of the jet relative to the tailplane position. Fig. 9b indicates

that 8e/SCj decreases with increasing Cj. The variation of downwash with wing incidence is more

important for stability and control considerations and the results are shown in Figs. 10a to 10d for r = 30 and 60 deg, and for various Cj- values. Ranges of values of (~e/O~)~= 0 are also

indicated on the diagrams, and are seen to be the same for the two different r values over the same

range of Cj for a given value of h/c. Since (0e/0c0~= 0 increases with Ca, it is not possible to assess a maximum, but for Cj = 4.0, (ae/O~)== 0 is well below 1.0 at the tailplane and on the extended chord-line, being 0.20 and 0.35 respectively. I t also appears that ae/~o~ increases as a increases, but ' this is only noticeable at the higher values of Cj, and for Cj = 4.0, ~ = 20 deg, Oe/Oa is still less than 0 .4 at the extended chord-line position.

The dependence of the downwash on the lift coefficient is shown in Fig. 11 as the variation of e/Cz, with C~, again for the two values o f h = 0 and 1.5c at l = 3.5c. For a plain wing, e/Cz is a constant for a given tailplane position, but Fig. 11 shows that this is not so for a wing with a jet-flap. The series of 'carpets' which appear on a complete e/C~ vs. Cz diagram (a, Cj and r varying), have been separated into the special cases of a = 0 deg, Cj = 2.0 and r = 30 deg in turn, each

with the other two parameters varying, e.g., for C# = 2.0, thecurves for c~ = 0 deg and ~ = 10 deg

with varying r and those for r = 30 deg and r = 60 deg with varying c, are shown, so that one

carpet appears. The curves are labelled with the value of the constant parameter on the right-hand

side, and the varying parameter on the left-hand side. For h/c = 0, it is seen that the majority of

values of e/Cz, lie between 1.2 and 1.6 deg in the CL range 1.0 to 8.0, and for h/c = 1.5, the

limits are approximately 0.7 and 1.1 deg. It may be mentioned that in an investigation of stability and control characteristics of jet-flapped aircraft, based on two-dimensional data, Taylor 4 used a

constant value for e/CL of 0.025 radn = 1.4 deg, which is of the correct order for small values

of h/c. It is interesting to note that for Cj = 2. O, e/Cz, increases with C~ if r is constant and e~ is increasing, but decreases as CL increases if a is constant and r is increasing.

The values of e/C~ for an infinite wing without a jet-flap, obtained by relSresehting the wing

by a vortex at the quarter-chord point, are 1.40 deg for h = 0 and 1.16 deg for h = 1.5c, the former

being a good approximation when compared with Fig. 11, but the latter value is appreciably greater than those obtained by the present theory, where the average value of e/CL is 0 .9 deg. For large X,

the value of e/Cz given by this approximate method is 1/(4rrX), i.e., it converges to the same value as given above in Section 2.

PART II

1. Vortex Representation of the Wing and Jet-Flap of Finite Span. As for the wing of infinite span, the pressure (and so velocity) difference across the wing and jet surfaces may be represented

by sheets of spanwise bound vorticity, but for the finite wing there is also a pressure gradient along the span, which gives rise to streamwise trailing vorticity on the wing and jet surfaces.

Maskell ~ has shown that the distribution of bound vorticity on a chordwise section of the finite wing and jet is approximately that on a two-dimensional wing and jet at an effective aerofoil

incidence (~e) and effective jet momentum coefficient C f ~. Thus the contribution to the downwash from the bound vorticity may be evaluated as in Part I, using Co, ~ and c~. in place of Cj and % but

in this case the resulting downwash field must be displaced through the total jet displacement,

i.e., the sum of t,he displacements due to the trailing vortices and to the bound vortices respectively.

In order to evaluate the downwash and displacement induced by the trailing vortices, the assumption

is made that the loading is concentrated on and near tO the wing, so that the trailing vortices arise

from one chordwise position, taken to be the centre of pressure of the effective infinke wing.

Since the loading is also assumed to be elliptic, the induced downwash may be evaluated in the

usual way 8. The representation of the finite wing is shown in Fig. 2. Let the origin of the non-dimensional

co-ordinates x, y, z (based on the chord being unity), be taken at the mid-span of the leading edge, and let UoF(x ) be the bound vorticity distribution on a section of the wing and jet. Then Maskell ~

has shown that UoF(x ) is approximately the vorticity distribution on a wing and jet of infinite span

at wing incidence dee, jet momentum coefficient Cj e and jet deflection ~-, where

Cje_ 1 cq (12) C j "r + dEe '

0~ i = dE - - dE e

Cz e (13) - 7rA + 2Qr '

\

CL being the lift on the finite wing.

Also Cz = aoe "r + a1~ % + 2CjdEi, (14)

where a0e and al~ are functions of Cj~ only, whose values are given in Table 1, for Cje = 0.5,

1.0, 2 .0 and 4.0. Hence

aoe'r + aledE "~ dEi-- 7rA +, ale

and

) ~ A d E - aoe'r dee - zrA ..J- ale

(15)

These values of Cje, ae and ~- may be used in Spence's solution to give UoF(x ) and the displacement

of the wake due to the bound vortices from equations (1) to (5), with dE and Cj replaced by ae and

Cj e respectively. The trailing vortices are assumed to arise from a lifting line at the centre of pressure of the

effective wing, i.e., at

x = x~ = (C~) _ b o e ' r + b l e % (16) e aoe "r +'aledEe '

where b0e and'bl~ are functions of Cj~ only, whose values are also given in Table 1 for Cje = 0-5,

1.0, 2 .0 and 4.0. For elliptic loading, the strength of the trailing vortices is

dir" - U0'~yy ,

4sC~ / ' - rrA sin0, y = - s c o s 0 , (17)

where

and s is the semi-span of the wing, based on the chord being unity.

* This relation assumes that dE~ = ½dE,o, which is not strictly true for large C 4 Values, but the approximation is acceptable in this context.

2. The Downwash due to the Bound Vorticity. The downwash at the point (X, Y, Z) due to

the bound vorticity on a finite wing and jet of span b = 2s, with elliptic loading and UoF(x ) as the bound vorticity distribution at the mid-span, is

- ( x - ~) (18)

to the first order in a and r.

In the plane of symmetry, Y = 0, the downwash becomes

wdx, o,z) = ~ ~ = 0

w h e r e y = - s cos 0, i.e.,

f~ UoF(x)(X - x) sin O s sin 0 dO 0 { ( X - x ) ~ + s ~cos ~ 0 + Z2} a/~dx'

Let

so that

T h e n

1 f ~ UoF(x)(X - .) w~(X,O,Z) = U~ ~=0 {(X 7 7)~ ¥ > ~- 2 ~ ) ~

S 2

( X - x) 2 + s ~ + Z ~'

k '2 = 1 - h a = ( X - x) 2 + Z ~ ( X - x ) ~ + s 2 + Z ' "

w ~ ( x , o , z ) = U~ ~=0 { (x - ~ -7 z~} k'~

2 s sin ~ 0 dO

I 1 - { ( X - x ) 2 + s 2 + z 2

k sin 2 0 dO (1 - h ~ sin s 0)a= dx

al~ dx .

l V o F ( x ) ( x - x ) l = 2~ ~=0 { (X Z ~ 7 k Z 2} ~ (E, - k'2K1) dx , (19)

where K 1 and E 1 are the complete elliptic integrals of the first and second kinds respectively, and

equation (19) has been derived using Art. 74 of Cayley's Elliptic Functions (Cambridge University Press, 2nd Edition, 1895).

Within the range (O,ov) of the variable x, the function . ( l / h ) (E 1 - k'ZK,) is found to have a

maximum value of approximately 1 at x = X, and values of approximately 0.5 at x = 0 and x = 2X.

Fur thermore, it tends {o zero as x tends to infinity. Thus for the range of x for which the function

F ( . ) ( X - .) (X - x) ~ + Z ~

has values differing significantly from zero and over which the major contribution to the integral

(19) occurs, that is, the range 0 < x < 2X*, the function ( l / k ) ( E 1 - k'2K1) is of the order 1,

and so the approximation is made that it is equal to 1 throughout the range of integration. I t would,

* The fact that the remaining part of the range of integration is infinite does not invalidate this argument, since for the numerical evaluation the part of the integral (19) over the range (l,co) of x is transformed into

F(x)(X - x) dx an integral over a finite range (O,rr) of~, say. The integrand is then ( Z ~ xy)2 7k Z~ dq5 ' which is still small

in the range of ~ corresponding to x > 2X.

8

perhaps, be more logical to adopt a value somewhat less than 1.0, but in view of the fact that the downwash due to the trailing vorticity is found to be much greater than that due to the bound vorticity, the approximation ( l / k ) ( E 1 - k'ZK1) = 1 is considered acceptable. The error in the total downwash arising from this approximation is estimated to be less than 12 per cent for large-aspect-ratio wings (A = 6, say).

Thus equation (19) becomes

1 [ ~ UoF(x ) (X- x) w,,(x,o,z) = ~ ~ = o (X -- . -5~7 an & , (20)

where

UoF(x) = Uof(X ) on the wing, 0 < x < 1

UoF(x ) = y(x) on the jet, 1 < x < oo

andf (x ) and y(x) are given by equations (2) and (5) with Cj - Cj~ and a - %, i.e., wB(X,O,Z) is the downwash due to the effective infinite wing, and may be evaluated as in Part I.

3. The Downwash due to the Trailing Vorticity. The downwash at the point (X, Y,Z) due to

the trailing vorticity of strength Uo(dP/dy) lying in the plane z = 0 is

- U° dyy {(x - X) ~ + (y - Y)~ + Ze} 3/~ dxdy ~v~(x,Y,Z)= ~ .=-~ ~=*~

= - ~ U ° - ~ y { ( y - Y-) 7 Z ~} 1 - { ( x , - X) ~--(y--Y)2+ ZZ}~/2 dy g = --g

where x~ is given in equation (16). F rom equation (17) we have that

dF 4sCz cos 0 U ° d y = U° ~rA s s i n 0 '

and so

" 47r=A U° J o s-sTn ? { ( - Y)~ + Z 2} X

I X - xp l 1 + { ( X - x,) ~ + Z 2 ~ (scos0 + y)2}~/2 s s i n 0 d 0

= -A~'C~ Uo f=c°sO(sc°sO+r) I _ X - x , 1 0 {(~ cos 0 + Y)~ + z~) 1 + { ( x x~)2 + 2 ~ 7 (, eos 0 + y)2)1/~ do

In the plane of symmetry Y = O, the downwash angle due to the trailing vorticity thus becomes

~ ( x , o , z )

C~, -- g o

(x - 1 ] 2Cz 1 " cos ~0 \ s , /

- 2 fo 0 + + ÷

2c~ I(~, 0 - - " ] T A

dO

(21)

" 9

where

and

Let

then

~: _ X - x~, Z ( 2 2 ) ~ ' ~ = s

if I ( ~ , 0 - - N o(COs2o+g2) 1 + ( c o s 2 0 + ~ 2 + ~2)~ a0

= N o 1 - ( l + ~ - s i n 2 O ) l + ( l + g i + ~ _ s i n ~ 0 ) l ~ 2 d0.

1 1 n = 1 + [2 and k 2 - 1 + ~2+ ~2, (23)

1 1 + 1 + ns in ~ 0 + (1 k ~sin 20) 1/2 + (1 + nsin 2 0 ) - ~ - h 2sin 20) ~1~ dO I(~,0 = U~ 0

- 2~ ~ + ~/(1 + n) ~ + 2¢kK~(k) + 2~nkn l (k ,~ )

1 I ~ 2 ~: - 2 . 1 . (1 + ~2)1~2 + ~ (1 + f~ + ~2)~/2 g~(k)

2 ~ : I - 7r (1 ~2 ~2)( 1 + ~2 + ~2)1/2 H~(k,n) ~ , (24)

where Hl(k,n ) is the complete elliptic integral of the third kind. A chart is given in Fig. 6 for the function I(~:,0 , so that eT may be evaluated from equation (21).

4. The Displacement of the Jet and Trailing Vortices. As in the case of the wing of infinite span, the displacement of the wake due to the bound vortices may be obtained from Fig. 3, using the relationship

z ~ ( x ) = ~, ~. ]o .~ + \ ~o~ L %. (2s)

The contribution due to the trailing vortices may be derived from the induced downwash at the wake, since

dz~ wi = Uo dx ' (26)

where

wi = induced downwash at the wake

zi = induced displacement of the wake.

The value ofw~ is given by equations (21) and (24) when X = x, Y = 0, Z = 0, i.e., ~ = (x - x2,)/s, = 0. Thus

2C~1 I 2f l w~(x) = Uo ~ ~ 1 + ~(1 + ~2)1/~ zq(ko) ,

where 1

k ° ~ - 1 + f2,

10

and so

~ ( ~ ) = ; 2 , 1 + (1 - ko~)~/~K~(ko) dx

Now

giving that

- wA x - x~ + -w (1 - ko2)l/ZK~(ho)~ dk o ,

~ = = 1 - ko 2 ko 2 ,

2_ [~o f r 3 1

dx (1 - - ko2) 112 ~ ( k o ) ~ o dkO . fi°

"/7"

$ (1 - ko2) *1~ K ~ ( k o ) dh o ko 2 (1 - ko2) 1/2

(27)

2, ;,~,, K#o) dko

= - ~ ! ~ 1 ko ~

= - ~ - o i z 0 h0~( 1 - k02sin20) 1/2

_2sf[' f' s i n 0 ( - t d t ) d 0 = - w ' cos0 ( 1 - #)al~t '

where t 2 = 1 - k02sin ~0

= ---2s do s in01 ( 1 - k ° 2 s i n 2 0 ) I I 2 k osinO + s i n 0 dO

and so equation (27) becomes

2s [ El(ko) ] zr ho + 1

Czb l x - x p 2 El(ko) 2 I z ~ ( x ) - 2wA s + - w k o w- . (28)

The function {z~(X)/Cz}(A/b) is shown in Fig. 5, as a function of ~ = ( X - x2o)/s. The complete downwash field w~(X,O,Z) + we(X,O,Z) obtained from equations (20) and (21)

must now be displaced through a distance z j (X) + z~(X), so that the wake is in.the correct position

relative to the tailplane, i.e., downwash at (X,O~Z + z j ( X ) + z i (X))

= . ~ ( x , o , z ) + ~ ( x , o , z ) . (29)

As for the wing of infinite span, the position of the tailplane is given in terms of l, the distance behind the wing leading edge, and h, the height above the extended chord-line, and so for the

evaluation of the downwash at the tailplane,

l h X = - + - a , (30a)

C C

I h Z = - H = -~ - - - za(X) - z~(X) . (30b) C £

11

The functions required in the analysis may be evaluated in the following order for given A, h/c, l/c, ~, ~ and effective jet momentum coefficient Cj~. :

(i) % from equation (15) using Table 1

(ii) Cj from equation (12)

(iii) CL from equation (13)

(iv) x~ from equation (16) using Table 1

(v) X from equation (30a)

(vi) ~ from equation (22)

(vii) zj(X) from Fig. 3

(viii) z~(~) from Fig. 5

(ix) Z from equation (30b)

(x) ~ from equation (22)

(xi) I(~,~) from Fig. 6

(xii) eT from equation (21)

(xiii) 3eB/~r, aez/Oa from Fig. 4

(xiv) eB from equation (11)

giving e = e~ + e;,.

It seems simpler to work in terms of the Cj~ as indicated above, and then to interpolate for a required Cj, as it is quicker to work through the procedure for three Cj~ values for which azj/Or, etc., are shown graphically, in order to find the variation of e with Cj, than to work through once with the Cj~ corresponding to the given Cj, haying to interpolate at each step.

For large X, it is possible to obtain a good approximation to the downwash far behind the wing without a jet-flap, by using the relation

2CL(1 s 2 ) e(X,O,O) = ~ - \ + ~ . (31)

However, in the present case of a wing with a jet-flap, the downwash far downstream is affected by the bound vortices in the wake, and these contribute a term [(CzA)/{4(wA + 2Cj)}](1/X), the next terms in the series being of order 1/X ~/~ and 1/X 2 with coefficients which are complicated functions of the Fourier-series coefficients _d o . . . _d,~ and B0 . . . B.. Thus the approximation

to the downwash for large X, if taken to the same degree of accuracy as that in equation (31), is too unwieldy to be useful.

5. Results. A few American experimental results for the downwash behind a jet-flapped wing of aspect ratio 8.4 were available, at the time of writing, and these are compared with the theoretical results in Fig. 12. More general results, calculated for a wing of aspect ratio 6.0, are given in Figs. 13 to 17.

In the American experiments, the wing was tested at zero incidence with Cj = 1- 6 and~ = 60 deg, and it is seen in Fig. 12 that the experimental downwash is greater than the theoretical values at the same tailplane positions. However, the experimental lift coefficient is given as 5.2, whereas

12

the theoretical value given by equations (12) to (15) is 4.2*, and so the calculations were repeated

using the experimental value of CL. The results obtained in this way are also shown i n Fig. 12,

and are seen to agree very well with the experimental curves.

The theoretical results for the downwash on the centre-line for the aspect-ratio-6 wing are

presented in Figs. 13 to 17 in a way similar to that used for the infinite wing. The variation of

downwash with tailplane position, Fig. 13, is given for the cases Cj~ = 2.0 (Cj = 2.7 to 3.0), -r = 30 and 60 deg, a = 0 and 10 deg. As for the infinite wing, the downwash varies only slightly

with the distance behind the wing when h = 2c. The corresponding downwash fields in the neigh- bourhood of the tailplane are shown in Fig. 14.

The variation of the downwash with the wing and jet parameters is again considered for the tailplane positions I = 3.5c, h = 1-5c, and on the extended chordline at I = 3.5c, h = 0. Fig. 15 shows the effect of varying the jet parameters, and it may be seen that the downwash at the tailplane

is about half that on the extended chord-line. The values of (~e/~a)~=o derived from Figs. 16a

to 16d are found to be appreciably larger than those obtained in Fig. 10 for the infinite wing, being

approximately constant at 0"70 for h = 0 and 0.35 for h = 1.5c for the various Cj values when

~- = 30 deg, and varying between 0.40 and 0.60 for h = 0 and between 0.15 and 0.30 for h = 1.5c

when ~- = 60 deg. Thus ae/ac~ decreases with increasing ~-, and also with increasing a and Cj, as

the graphs show, although e itself increases. However, even for the lowest values of a, Cj and ~-

considered (~ = - 1 0 deg, Cj = 0.5, ~ = 30 deg), ~e/ac~ is still well below 1.0 at the tailplane

position.

The carpet plots of e/Cz against Cz (Fig. 17), are similar in shape to those for the infinite wing

(Fig. 11), but the values of e/CL are much greater, being between 3 and 6.5 deg for h = 0 or

between 1.5 and 3.5 deg for h = 1.5c, in the C~ range 1 to 8, as compared with the ranges 1.2

to 1.6 deg for h = 0 or 0"7 to 1.1 deg for h = 1-5c obtained from Fig. 11. The carpet plots of

e/Cz~ for a wing without a jet-flap are also shown in Fig. 17, the lift being equal to that on the

jet-flapped wing in each case, and the values of e/C~ being obtained from the Data Sheets:

Aircraft 08.01.04 and 08.01.02 of the Royal Aeronautical Society. I t is seen that the data sheets

underestimate e/Cz for the position on the extended-wing chord-line, but at the tailplane height

of 1.5c, the agreement is good for some specific combinations of values of Cj, "r and a, e.g.,

Cj = 2.0, ~ = 10 deg, ~- = 30 and 60 deg. However, it does not seem possible to estimate the possible error in using the results from the data sheets for a given C~, because the parameters C j, ~r and ~ affect the downwash and lift coefficient in different ways.

Conclusions. The downwash behind jet-flapped wings has been evaluated, using the thin- aerofoil theories given by Spence 1 and Maskell ~, for wings of infinite and finite span respectively,

to obtain 'the strength and position of the vortex sheets which represent the wing and the jet. Thus the basic assumptions are that the wing incidence and jet deflection are small, and that the finite wing is elliptically loaded. It is also assumed that the trailing vorticity on the finite wing may

be represented by a vortex sheet arising from one chordwise position on the wing. The resulting

downwash fields have then been displaced so that the jet is in the correct position relative to the tailplane.

* This difference is mainly due to the use of a blown flap on the experimental model, rather than a jet-flap. The downwash will also be different, and for the blown flap it could be calculated using the results of Ref. 7 in place of Ref. 1, i.e., Ozj/~-r, ~ez/~'r , aoe and boe must be changed. However, for the present comparison, it is sufficient to take the difference as being due only to the increase in lift.

13

The agreement with the few experimental results available is quite good, especially if the experi- mental C~ value (which is appreciably greater than the theoretical valuee), is used in the calculations. The variations of the downwash with tailplane position, wing incidence and jet parameters follow similar patterns for the infinite and finite wings, although the effect of reducing the aspect ratio

from infinity to 6.0 is to increase the downwash by factors ranging between 2 and 4 for the con- figurations considered. Of the derivatives with respect to the wing and jet parameters, ~ e / ~ is the

most imi~ortant from the point of view of aircraft stability, and it appears to be well below the

critical value of 1.0 for tailplane positions above the extended-wing chord-line. One major difference

between the infinite and finite wings is noticeable here, in that ~ e / ~ increases with increasing

c~ and C j, and is nearly constant with increasing ~- for the infinite wing, but it decreases as ~, Ca and 7

increase for the wing of aspect ratio 6.0. It is found that e/Cz~ is not a constant for a given tailplane

position, when considered as a function of C5, due to the fact that there are three parameters

(% Cj and ~-) which may be varied to obtain any given C~, and these affect the downwash in different ways.

No consideration has been given in thi s report to the effect of the rolling-up of the vortex sheets in the wake when the aspect ratio is finite, but some rough calculations have been made based on a completely rolled-up trailing vortex system, and those indicate that the downwash on the centre-line will be increased somewhat at tailplane positions near the jet when rolling-up takes place.

* The theoretical value for a blown flap is also greater than that for a jet-flap, for the same .angle of incidence of the main wing.

14

LIST OF SYMBOLS

l

Note. 'Infinite' refers to the infinite-span wing in Part I and to the effective infinite wing in Part II when the symbol occurs inboth Part I and Part II.

aoe

al~

hoe

A

b £

Cj

CL

el(k)

f(x) f (x )

h

k, ko, n

K~(k)

l

S

8o gO B

fog

gO

X2a

(.x. Y,z)

~CzlS'r for effective infinite wing

8Czl~c~ for effective infinite wing

8C~/~- for effective infinite wing

~C,~/O~ for effective infinite wing

Aspect ratio

Fourier-series coefficients

Wing span, based on the chord being unity

Wing chord

Jet momentum coefficient

Effective jet momentum coefficient

Lift coefficient

Complete elliptic integral of the second kind

fl~- + f2o~ (Non-dimensional vorticity on infinite wing)

Non-dimensional bound vorticity on finite wing and jet

Height of tailplane }-chord point above the extended wing chord-line

Function introduced in equation (21)

Variables for the elliptic integrals

Complete elliptic integral of the first kind

Distance of tailplane i-chord point behind the wing leading edge

Wing semi-span, based on the chord being unity

Mainstream velocity

Downwash velocity due to bound vorticity on effective wing and jet

Downwash velocity due to trailing vorticity

Induced velocity due to trailing vorticity at the wake

Complete downwash at the tailplane

Non-dimensional co-ordinates, based on the chord of the wing being unity

Distance of centre of pressure of effective infinite wing behind wing leading edge

Displacement of infinite jet

Induced displacement of the trailing vortex sheet

Co-ordinates of point at which downwash is to be evaluated

Incidence of wing

15

(~

r(x)

UoF(y) e = w l U o

'7"

L I S T OF SYMBOLS--contin'ued

Effective incidence of wing

Uo(71-c + yzo~) Vorticity on infinite jet

Total bound vorticity on wing and jet at any spanwise station

Angle of downwash, with corresponding subscripts B, T, downwash velocity

Complete elliptic integral of third kind

Angle of ejection of the jet relative to wing chord

'Co-ordinates' introduced in equation (22)

as for the

i6

No. Author I D . A . Spence . . . .

2 E . C . MaskeU and D. A. Spence

3 H. Multhopp . . . . . .

4 A . S . Taylor . . . . . .

5 R . D . Vogler and T. R. Turner ..

6 J . R . Spreiter and A. H. Sacks ..

7 D . A . Spence . . . . . . .

REFERENCES

Title, etc. . . A treatment of the jet-flap by thin aerofoil theory.

Proc. Roy. Soc. (A). Vol. 238. pp. 46 to 48. 1956.

•. A theory of the jet-flap in three-dimensions.

R.A.E. Report (To be published.)

. . The calculation of downwash behind vdggs.

L.F.F. Vol. 15. 1938.

.. An examination of some longitudinal stability and controI problems of jet-flap aircraft with particular reference to the use of jet thrust and jet-flap deflection controls.

R.A.E. Report (To be published.)

. . Wind-tunnel investigation at low speeds to determine flow-field characteristics and ground interference on a model with jet-augmented flaps•

N.A.C.A. Tech. Note 4116. September, 1957.

.. 'The rolling-up of the trailing vortex sheet and its effect on the downwash behind wings.

J.Ae.Sd. Vol. 18. No. 1. 1951.

•. The lift on a thin aerofoil with a jet-augmented flap.

Aero. Quart. Vol. IX. August, 1958.

T A B L E 1

CJ~

0.5

1.0

2 .0

4 .0

2.707

4.026

6.135

9.638

7.588

8.605

10.405

13.609 I

( cq b°e ~ ~T ].e

1.485

2.315

3.745

6.307

bl~ = \--~--],

1.898

2.097

2.396

2.836

17

(7ss24) B

Lt~, x .~ I-

• g, .)

T-- H

U.o u.o f ( , ) 1 ,,@_- - ~ U.o,-';@

/ ~ • P 'C~ , : )

t Z

FIG. la. Section of infinite-span wing, showing jet and tailplane positions.

Fro. lb. Vorticity distributions and tailplane position as used in calculations.

CO

U.Q L )(" ~f

d

X p

LLo ' U.,:, F ( x . ) d' ~)ip

a t i

T-" cLP H

Uo~- 1 J

3¢

Fm. 2a. Section of finite-span wing, showing jet and tailplane position.

FIG. 2b. Vorticity distributions and tailplane position on centre-line of wing.

I 3 '0

?-'5

i ',- . /

:, {

/ /

,sf.,. 3 ' q

C ~ =

o

f

/---'-C4~,.o ..._..---.-'-

S ×

Z-:r

1"5

1.0

0"5

/ /

/

I /

J / -

/ l • " I

j ~ o---~w---

I 9"0

3 ~ 5

z f~z~h

FIO. 3. Charts for determination of jet displacement (infinite wing), or jet dispIacement due to bound vorticity (finite wing).

19 {78824) B*

FIG. 4a.

~P-B

C~e = o.5

Charts for determination of downwash of infinite wing (Cj = Cj ~ = 0.5), or downwash due to bound vorticity of finite wing (Cje = 0.5).

3.0

Izl

2'0

I'C

3"0

I I

)'12

C3e = I 'O

i

5 ' 0

Fie. 4b. Charts for determination of downwash of infinite wing (Ca = Cje = 1.0), or downwasli due to bound vorticity of finite wing (Cje = 1.0).

20

' ---I C T e = :~,o

FIG; 4c. Charts for determination of downwash of infinite wing (Cj = C j e = 2-0), or downwash due to bound vorticit)r of finite wing (Cje = 2" 0).

FIG. 4d.

C~e = 4 . 0 3.o ~ 1 I

o

J /

~,L4-

o, ls

- ~ o.~

N 5-0

Charts for determination of downwash of infinite wing (Cj = Cje = 4.0), or downwash due to bound vorticity of finite wing (Cje = 4.0).

21

bO t-O

0"4

O.Z,

O-E

o-i I

/ 0

0 0-I O.Z

J L

f J

J / J

I '0 I'1 I'~. 1'.3.

/

1"4

I / J f

/ /

O'3 )'~ 0"5 0'6 0"7

J

1"5 /

J J

" /

J 1'7 I

0"~ 0"9 I'O

Fro. 5. Chart for determination of induced displacement of jet and wake (finite wing).

b

~I ~I °~1o. °I °1 ~1

FIG. 6. Chart for determination of downwash due to trailing vortices (finite wing).

23

I:~1= 21zl b

e~ = {2c~/(~A)} z(~, ~).

(78824) C

bo

,o ~

5 °

lff

5 ~

' ~ : 3 d c:4. : ~ -°

~.o -. ~ . . . . . . ~ .

-a.ol

3 0 '~-0 5 0 ~-/c

~:3d

e~: I0 ~

CL:5-O~

; " O -

3 "~£c = O,

5'

d 3.0 ~ . 0

(b)

I-O ,~,

~ - O - ~ 5

~ : 6d

~: o °

CL: G'~.i

"~: 60 °

~ : I0 °

3.0 ÷-o so ~/c 3.0 4--o so % (c) (d)

FIGS. 7a to 7d. Variation of downwash with tailplane position (Aspect ratio = oo, Cj = 2 .0 , ~- = 30 and 60 deg,

= 0 and 10 deg).

&O

C.

I '0

3.O 4-.O

?-'C

I.O

5'0

"C:50°~ ~ . : O °, C-,.: ~,-EI

3 '0

G °

~-.o "8"/0 5-0

"~= GO~ o~.:O°~ C L : 6 . 4 - E

o

5"0

---<.

~-o ~/c 5"0

~t:lO°~ C=: 5'o3

3"0 4 , 0 ~/c. 5"0

~ : G O ° ~ a ~ , : l O ~ CL: 8"~-

FIG. 8. D o w n w a s h field in region of tailplane (Aspect ratio = oo, Cj = 2-0 , -r = 30 and 60 deg,

= 0 and 10 deg).

,o'

E' i&

' TY J

J /

° o ~ , 0 ° ~o° 3o°

. /

~0 ° 50 °

(O) VARIATION WITH JET DEFLECTION C~.: 2 . 0

i

/ - - ~ c = t.s

/ I

~.0 3"0 ~'0 C~ 5.O

,o° I

6 ,

5 °

/ f

o o I.O

6 0 °

( b ) VARIATION WITH JET MOMENTUM "~ = 30 °

FIGS. 9a and 9b. Variation of downwash with jet parameters (Aspect ratio = oo, ~ = 0 deg, l/c = 3 "5,

h/c = 0 and 1.5).

26

hO

IS °

O" Z.O "~ (~ .~1 < 0 " 3 5 / "" ~=o /

i0 o f . , 1

- i 0 ° 0 ° i 0 Q

(a) g : o , - = = ~ o °

. C ~ : 4 - ' 0 / .,.~.~ .--~ C,:r : Z-'O

/

,,.,~.. C~r = 1-0

~ - 0 .~ ~L

o . + s < (~"+"'~ < o - a o ~ . ~ - c~=+.o

/

•

"10 Q 0 I 0 ° 2 0 °

Cb) + : , .s, '+,: : 3o °

Fins. 10a and 10b. Variation of downwash with wing incidence (Aspect ratio= oo, l/c = 3" 5, h/c = O and l .5 , -r = 30deg).

/ /

f J / I

/ J

/

~'~1 i , c... = ÷ . o

IS" - ~.=o , / J J " c'T= ?-'0

/

o

Cc) + : o , m : 6 o °

r - I . ~ I

1 5 °

- L O ©

I0 ° ~ 0 °

c~= 4-.o tOO _ O , i S < L . ~ ) < o . ~ . o ..----~

~.=o° / / - / - " " / ~ C~ = ?.'0

/ ~ ~ . _ ~ . / / C~= i .O

/

0 tO ° P-O ° ~k

(d) ~ : . , s . ,= : 6o °

Fins. 10c and 10d. Variation of downwash with wing incidence (Aspect rat io= o~, l / c = 3" 5, h / c = O a n d l ' 5 , -r= 60deg).

/

1.4 ° CK- - " ~ I 1'4 c .,,,..=? . ~ . [ .o cs:l.o ~ - " ~ . ~ 6o° .

Cj : 2"0 -

1.0 ~

o.~"

O.G °

0.~. °

0 '2 ° ~=o o, I~oo

t.o ~

o.~

O'G

0"~

0 ' ~

0 l Z 3 4 5 G 7 8 C L 0 I

[ ! ~ I .g :30 ~ CL CT: 2 .0 - -

~ . ~ , : ,o ~ ~ ~ ° , - ~ - ~ - - i ~ `,.°'

/

C:}-: ?. '0 ~ ~ : 0

0'8

0",

o.4'

O.Z'

3 4 5 6 7 B CL o I ?~

1 "C : 30° ~ ~ : 0

I I I 4 5 G 7 8 CL

CL Z ~.~3

O , g °

0,~ °

o.Z

o..~ a

° o I "~ 3

~ , . o ~ : ~ ~ ~.___c,o' ! ~ ~< - ~ , ~d.:O ~ Cj,:4"O"

~ 0 o O.ro ~

0"4 ° ~ 0"4'

.~ : o' , ~ : ,.~ o.~" c~: ~ . o , ~ o , .5 -0.2' "~: so', ~ o , .s

I I I ~ I I I I 5 6 7 8 CL. o I Z 3 .4- 5 6 7 B CL. o I ~ 3 4 5 G 7 8 C L

Fie. 11. Variation of e/C5 with Cz, for a, Q1 and ~- kept constant in turn (Aspect ratio = oo, l/c = 3"5, h/c = 0 and 1.5).

28

bO ',,.0

" x P ERIME N TITj~t TH EO RY

JET DISPLACEMENT

IO ~

- ~ - I

\ k \ '~/C = 3 ' 0

, \ . ' . ~ / T H E O R E T I C A L VALUES B~SEb ON EXPERIMENTAL C L

", , 'x. I I ~ \ ~ E X P E R I M E N T A L VALUES

"% \~'~,AT 0'~5 ~, SEMI~PAN.

THEORE VALUES ON ~..

, ~ ~-

EXPERIhENT-] [-THEORY

J TET ID@~¢ PLACEMENT I0 °

-?, -I

~/c 40

\- oE,ol I "LOE BS O

\

7" THEORETICAL VALUES ON ~_~

Z 3

FIG. 12. Comparison between American experimental data and theoretical results (Aspect ratio = 8.4,

= 0 d e g , C j = 1.6, ~ - - 60 deg. Experimental C5 = 5 .2 ; Theoretical C~ = 4. 20).

° ZC

1 5

i o °

% = B O °

~ = 0 °

CL= Z'G7 Cj = Z'7

5 °

zd

IS '

, O ° ,

~ = o \ % '=60° d,.= 0 ° Cu= 5 " 5 5

'5° ~ GX = ~"7

I 'O "~

3.o 4 .o "6/c s.o .3.o 4 .o ~ / c s.o

(a) (b) g:o\

% = 3 0 ° ~ "C-- GO ° ~=O" ~ = I0° , ~ cA.= lO ° C ~ CL= 4 - ' Z 5 CLING'93

I ' 0 ~

S °

~ .O

1.0 ,~

2." 0 ~ ~ ~ ' - - - - - - - .

5-0 Lt-.o "6/c. 5.0 3 .0 &.o '6/c 5-o

( c ) (d)

FIGS. 13a to 13d. Variation of downwash with tailplane position (Aspect ratio = 6.0, Cj = 2 .7 to 3.0, ~- = 30 and 60 deg, c~ =-0 and 10 deg).

G o O

2.'0 ~ ~-.-C ¢ , . ~ -f,.

4--0 ~O S'O

"1 : ; :60° eL=O° G L = 5 . 3 5

S'O 4-.0 "&,/c 5 '0 5.o

"~ : 5 0 ° e4,:O°> C L : & . G 7

~-'0

0 3"0

o 4..0 -~/¢ s-o s.o 4.0 % s.o

" ~ : 3 0 % ~k= I0°~ Ct..:q-.~.5 m_.=GO°~ ok=.lO°, 0 L : 6 . ~ 3

FIG. 14. Downwash field in region of tailplane (Aspect ratio = 6.0, Cj = 2 . 7 to 3" 0, 7 = 30 and 60 deg, a = 0 and 10 deg).

2.0 °

15 °,

io o

/ J

-C =°

,• / l j 5 ° /

O I0 ° 2.0 ° 3 0 ° . 4-0 ° 5 0 ° "c

(Q) VARIATION WITH J E T DEFLECTION C~. = 2"7

/

/ ~ I s c

15 °

,, L / J

s ° / /

Y o

0

J

/

/ /

c

I / J

I '0 Z..O 3"0 4 , 0 C~r 5 '0

..Ik - - = . I.g G

( b ) VARIATION WITH JET M O M E N T U M "12 = 3 0 '~

FIGS. 15a and 15b. Variation of downwash with jet parameters (Aspect ratio = 6.0, a = 0 deg, l/c = 3 .5 ,

h/c = 0 and 1.5).

G o °

/ /

"lO° 0

(a)r = +o °, +: o

20 D / ~

/ f ~ / (+~ ~- o-+o

I0 °

"I0° 0 " I0 ° a~. 200

(b)l : = 3o ° , + : , . s

FIGs. 16a and 16b. Variation of d o w n w a s h wi th w i n g inc idence (Aspect ratio = 6 . 0 , l/c = 3"5,

h/c = 0 and 1" 5, ~- = 30 deg).

/ /

/ /

f , . /

/ + o . + + ° < o . °

- I 0 ° • 0 °

(c) +:+o °, %: o

Io ° ¢£, 2 - 0 °

/

_jo o o o io o ~ ?.o °

(d) +C--6o +, ~ = ,.S FITS. 16c and 16d. Variation of downwash with wing

incidence (Aspect ratio = 6.0, l/c = 3.5, h/c = 0 and 1.5, -r = 60 deg).

C~.

6 =

5 °

5 °

Z °

i °

0

CL S o

#

~o

Z °

O 0

"x..

d-= 0~; " ~ ' : 0

I I

CU~,VE5 FOP, % = SO =

AND C: = I'O ,~,P.E

ALMOST COINCIDENT

"WITH TI'~0SE FOR % = 6 0 ~ AND C~ = ~.'0

7 ~. C L

, : / ,=0 ¢', --~I=I 'S

I ?- 3 ~ 5

~ = 6 o °

7 8 C u

CL

6 = ~ ~-~ 'I

s° A_ ..-~- ~" ~---'--" ~ = 3o °

-. ~_ ,~= Go° ÷o

f f ~= 0 3°

l ~ C z = ?. '0, ~ = 0 i o

t I I I E 3 ~ 5 6 7 8 CL O l

0 PP~ESENT VALUES GIVEN BY TI-~EOP.Y - - -1~- - - DATA 5EtEET5

OF P.OY, AER.O. SOC. E E

5 { y

Cz~, c~,, ~ ~ . . . . ~ . ~ . 5" ~ - 5 °

~ " ~ ~ =,¢.:o~=10 = d . : 2 . o o __ I ° I*

0 i ,2. 5 ~ 5 G 7 8 C u 0 l Z 5 ~ 5' G 7CL8

E

• 5 D ~ ~ . . . . %c~ = ~.. o

ok= 0 d~-- 2.0 °

I x=3o ~, & o -X-=

I I I 2. 3 .4- 5 6 7 8C.

FIG. 17. Variat ion of e/C z with C~;, for o~, C j and ? kept constant in t u rn

(Aspect ratio = 6 . 0 , I/c = 3 .5 , h/c = 0 a n d 1 . 5 ) .

(78824) Wt. 54]82]0 K.5 2[61 Hw.

32

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