TURBULENT BOUNDARY LAYERS ON DELTA WINGS AT ZERO LIFT
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U.D.C. No. 533.693.3 : 532.526.14. : 533.6.011.5
C F. I'll c. 6'96
Ara.rch_ l y63
TURBULENT BOUNDARY LAYERS ON DELTA WINGS AT ZERO LIFT
by
J. C. Cooke, D.Sc.
It is found that, for turbulent flow at Mach number 2 ovor a thin deli
wing at zero lift, the effect of pressure gradient on the boundary layer Is
negligible; thus boundary layer calculations allowing for convergence and
divergence of streamlines are simplified. When these are done it is found
that, except near the centre line, where streamline convergence causes extr?.
thickening towards the trailing edge, the momentum thickness is nearly the
same as it would be for flow over a flat plate of the same planform. This
enables the boundary layer pressure drag and the skin friction drag to be
determined simply. It is found that the pressure drag may be neglected coni
pared with the total drag, whilst the skin friction is the same as that of 8
flat plate of the same planform.
Replaces R.A.E. Tech. Note No. Aero. 2878  A.R.C. 24,884
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LIST OF CONTENTS
I INTRODUCTION
2 THE WINGS CONCERNED
3 EFFECT OF PRESSURE GRADIENT IN TWO DIMCNSIONAL CALCULATIONS
4. THE SHAPE OF THE EXTERNAL STREAMLINES
5 THE EFFECT OF STREAMLINE CONVERGENCE OR DIVERGENCE
6 SOME ACTUAL MAGNITUDES IN A TYPICAL CASE
7 THE EFFECT ON TF PRESSURE DISTRIBUTION
8 THE BOUNDARY LAYER PRESSURE DRAG
9 THE SKIN FRICTION DRAG
10 CONCLUSIONS
LIST OF SYMBOLS
LIST OF REFERENCES
APPENDIX I  Determination of L14 and Aop
TABLES 14
ILLUSTRATIONS  Figs15
DETACHABLE ABSTRACT CARDS
LIST OP TABLES
Table
I  Values of oonstants in Spence's equation. Zero heat
transfer
2  Boundary layer thicknesses at the trailing edge 1C
3  Value of K(r1) for Z = 0.8 11
/+  Drag ooeffioients for the wing tested by Firmin 13
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LIST OF ILLUSTRATIONS
Fig.
Values of (0/0)1 by two dimensional caloulations
Calculated external streamlines Nos,15
2
Values of (0/0)1,2 along streamlines 15
3
Isobars of Aop for R = 107, (3s/c = 0.577
4
Dop where y/s = 0.225, by simple wave and slender wing theories
5
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I INTRODUCTION
In an attempt to assess the effect of the boundary layer on the drag of
slender wings at zero lift, turbulent boundary layer calculations are wade for
a certain delta wing which has been tested at Bedford, using the measured
pressure distribution and, in oases where this does not give enough informa
tion, using also calculated cross velocity components.
Firstly a simple calculation is made by Spence's method1 assuming the
flow to be twodimensional along a series of chordwise sections. This clearly
shows that the pressure gradients on the wing are so small that the boundary
layer (calculated on the very simple twodimensional basis) behaves almost
exactly as though the pressure gradient were zero everywhere, that is, as
though the flow were over a flat plate. This simplifies the subsequent work
since the pressure gradients can be ignored leaving it possible to oono;ntrate
on the effect of diverging or converging streamlines. Tho wing concern.)d had
an 11% thickness chord ratio. For thinner wings one may expect this conalusiomz
to be even more justified.
.A second set of calculations is then made. It consists of two parts 
firstly the determination of the external streamlines, and secondly the oalcul3
tion of boundary layer momentum thickness along those streamlines, allowing for
convergence but not for pressure gradient. It is found that, except for stroara
lines very near to the centre line, the momentum thickness is still very close
to what it would have been on the flat plate assumption. Near the centre line
convergence of the streamlines causes considerable thickoning of the boundary
layer towards the rear, but this effect decreases very rapidly as we go
outboard.
The next step, therefore, is to ignore the effect of convergence and to
assume that the momentum thickness and displacement thickness over the wing arc:
the same as over a flat plate. Thus a displacement surface is very simply
obtained and the effect of this on the velocity potential 0 is expressed in
terms of an added function 1O; thus Lop, the increase in the pressure
ooeffieient, can be calculated, and isobars of Aep may be plotted.
Finally by integration over the surface of the wing the boundary layer
normal pressure drag coefficient is found. This drag is positive in the first
example under consideration but it is very small. In fact its value at Mach
number 2 and Reynolds number 107 is 0.00008. At higher Reynolds numbers it
will of course be less than this. The calculated inviscid wave drag coefficiert
is 0.00821; thus the boundary layer pressure drag is 15 of the invisoid wave
drag. This indeed may be an overestimate, since it assumes a displacement
thickness which, as has already been pointed out, is tbo'small at the rear near
to the centre line. This increased thickness hero will give an increased op
which, being on backward facing surfaces, c?,rill reducetho' drag. This effect,
however, only occurs over a narrow band and so the reduction will be small. It
seems unlikely to be sufficient in this example to give negative drag, though
this could possibly happen in other examples.
As already pointed out, the wing on which these calculations were made had
a maximum thickness chord ratio of over 115%. For thinner wings one might expect
the flow to be even closer to that over a flat plate. The same line of approach
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may be used for other planforms besides deltas, though the analysis in such
oases would be more difficult.
'ormulae are given which enable L1op to be determined at any point of any
slender thin delta wing at zero lift at any Mach number or Reynolds number.
Thus by integration the boundary layer pressure drag of the wing can be calcu
lated. The skin friction will be the same as that over a flat plate, or
possibly slightly less in the present example owing to the behaviour of the
momentum thickness near to the centre line. The main conclusion, however, is
that the boundary layer pressure drag is small and may probably be neglected
at full scale. A second example was considered later and for this there is a
reduction in pressure drag which amounts to 3 at R = 107, due to the thickness
of the boundary layer.
The flow is supposed to be compressible and everywhere turbulent. If
there are areas of both laminar and turbulent flow the calculation of 6o pis
more difficult; another complication is the sudden decrease in displacement
thickness which occurs at transition owing to the sudden drop in the value of
the shape factor H which takes place, whilst the momentum thickness remains
continoous2. Since at full scale the flow is likely to be turbulent over most
of the wing we do not consider here the case in which it is partly laminar.
The work done here only applies to wings at zero lift. At higher
incidences it seems probable that the method of simplification given here would
not be possible; it may be so, however, if the flow is attached along the
leading edges of a cambered wing at a low lift coefficient.
There seems to be no check on this theory by experiment as yet. This
would be a difficult undertaking, but accurate measurement of a few boundary
layer profiles on the surface of the wing near to the trailing edge would be
of groat help.
2 THE WINGS CONCERNED
Two examples were used. These were both of delta plan form and had
equations
12 }3 L.
? 2 ~4  10 a + 10 ~o  5 ~c/ + (20fl(i_ Is 1
a/c = 1/3
v = o? 01 02 ,
known as the "Lord V" wing, which was tested at Bedford, and
2 (io ) 3 ~4)
Z = + 2 142 + 116 660 + 852  350
/ \%j
a/c = 1/4 ,
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which was tested at Farnborough by Firmin3, who named it Wing 3. This wing
is such that at the trailing edge
S10) V 16 ,
0 103
where S(x/c) is the cross sectional area and v is the total volume.
Here s is the semispan at the trailing edge and c the root chorl.
In the case of the Lord V wing agreement between calculations of pressor:
distribution by slender wing theory was good. This did not apply to tie
second wing and so calculations were made for it by Firmin by linear wing
theory. He found that this theory gave fair agreement with his experiments.
This gives ground for the hope that the calculated values of cb and ? of
necessity used in Section 5 below may not be too much in error.' The socond
wing has large, backwards facing slopes at the rear and thus cannot be tonsideir.d
"slender".
3 EFFECT OF PRESSURE GRADIENT IN WODIMENSIONAL CI`LCULATIONS
A cartesian coordinate system is used, the median piano of the wing
being z = 0, with the xaxis along the centre line. The equation of the wins
surface is z = z(x,y) as In Section 2, and dz/ax and dz/ay are supposec sm al1.
We choose the method of Spence 1. In the absence of a shock the c quat ort
for the momentum thickness 6 in a turbulent boundary layer may be written
(?) 1+n ue B+n T D 1 E B
\ j ` ?l Rn n+1 C f (TT_P m\ ufx\
o u T n ~T ) o " d ; + co start
oooaaa"' ~ ` J
In this equation the subscripts e, oo and m refer to values at the edge or
the boundary layer, at infinity and at a certain "moan" position respectively.
R, the Reynolds number, is equal to u00 o/ co .
Depending on the ranges of R0(= uee/ve) concerned (which overlap) n may
take the values 1a., 5 or higher values. We give in Table 1 the values 6t' the
constants for zero heat transfer when n = L. and n = 5.
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TABLE 1
Values of constants in Silence's equation. Zero heat transfer
n=Z n=5
C
0.0128
0.00885
n+1
n C
0.0160
0.0106
B
4125
40
D
1.735
1.665
E
1.332
1.31f3
F
0.778
0.822
H
2.5(1+0.178M2).1
T/T0
i+0128M 2
Range of Re
1005000
50050,000
The value n = 5 was chosen for the first calculations. Taking the
measured pressure distribution at various values of y/s for the first wing at
Mach number 2 and Reynolds number 107, based on root chord, the solutions in
Fig.1 wore obtained (circles).
If there had been no pressure gradient, so that ue = od Te = Toe AM as on a flat plate, equation (1) on integration would have reduced to
1+ 1 1 g
n
Cep = n+1 C R n  C1 + 0 1281.1 c 
c ( ; , (2)
n 00 s
assuming 6 to vanish at the leading edge, which will be the case if this edge
is sharp.
For R = 107,
or, forn=t+
M = 2, n = 5 equation (2) becomes
Cep1.25 \
= 0.000206
~c isl%
0.000300 \~ 
;s
(3)
(4)
1'2
(81 obtained by equation (3) is plotted for the first example as a full
c
line in Fig.1 for comparison with the results with pressure gradient. As can
be seen the result is scarcely distinguishable from that obtained by a full
solution of equation (1). Equation (1+) gives results virtually coincident with
those of equation(3). The same conclusions apply to the second example.
7
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Thus we may say that the measured pressure gradient of the wing at zaro
lift is so small as to be negligible in boundary layer calculations. This day,
not always be true. The equation from which (1) is derived is
dx + $ a au r2 + HM2J Tut
\\ // Pe e
and the effect of the pressure gradient lies in the second term. (It must zil.so
affect T to some extent but this is generally ignored.) Now from Table 1
2+ HM2 = 3.50.555M2
and this vanishes when Iii = 2.51, which is not very far away from the value
M = 2 used in the calculations.. At any rate we have shown that the pressure
gradient has very little effect in our examples and we shall ignore it from new
onwards.
THE SHAPE OF THE EXTERNAL STREAMLINES
The streamlines are calculated from the equation
y ve
dx  u '
e
where ue and ve are the x and y components of the external velocity. )nly the
value of ue can be obtained from the measured pressure distribution ani so v
was found by a slender thin wing calculation for the given wing. The 3olutio a
of equation (5) is straightforward but involves some interpolation and
iteration. If v6 is calculated for a few values of y/s near to the pa:rrtieuia:
one concerned the interpolation can be done graphically. Once y is found,
r
(which is required in later calculations) may also be found by interpolation.
Some of the streamlines for the first example are shown in Fig.2. The7
diverge near to the leading edge but converge later. However, the convergon
is very slight except near to the centre line. This oonvorgence is very mur.;'l
less in the second example.
5 THE EFFECT OF STREAMLINE CONVERGENCE OR DIVERGENCE
According.to.the axisymmetric analogy .the boundary layer along any
streamline on the wing z = z(x,y) behaves like that over an axially symmetrio
body of radius r, whore r is given by
Ue as log r2 U2)
au? + av
2
(dx ay
assuming that az/ax and az/ay are small. Here we have written U2 T u?+v? arc s
represents distance measured along a streamline.
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T TS 
Ue or ave
ay
(7)
ignoring the velocity gradient aUe/as and ignoring also aue/ax compared with
ave/ay in accordance with the usual slender body theory. In any case aue/ax
is approximately equal to aUe/as which we have already decided to ignore.
If the external perturbation potential is o0' and we write Ue = u00,
equation (7) becomes
r as = ?yy
(8)
Now Spence1 gives the form of his equation for an axisymmetric body. It
is the same as equation (1) except that rff+1/n is to be inserted in the left
hand side ant also inside the integral on the right hand side. In using the
axisymmetric analogy we must follow a streamline and hence d(x/c) should be
.1. As we are ignoring the
replaced by d(s/c). we must also replace u e by U(I
pressure gradient we shall write Ue = u., Te  T U = LLi It is more conOOP 00
venient to differentiate the equation. UsinG the version n = 5 in Table I and
writing 9 = (0/c)1`2 we find
d0 +1.211 Or
d s c r a s 7c7
0.0106 R0.2 (1 + 0.1281.,40.822
01) P
ors for P: oo = 2, R = 107, using equation (8)
d x c
(9)
where we have replaced s/c by x/c, since the streamlines are nearly parallel to
the xaxis.
If 0yy = 0 this equation has equation (3) as its solution, as was to be
expected. Thus the effect of convergence or divergence of the streamlines is
expressed by the term 1.2c Oyy ? in equation (9).
The solutions of equation (9) for the first example are shown in Fig3
for various streamlines, numbered I to 5 in Fig.2, together with values from
equation (3).
The main feature of the curves in Fig.3 is that the solutions by equation
(9) and the flat plate solution run very near to each other, except near to the
centre line, where the error in 0 rises to about 50j,. This is, however,
confined to an area very near to the centre line. At other locations the
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initial divergence reduces the value of 8 slightly and the convergence whae:)
occurs downstream has little effect on ?. Consequently in calculatir the
effect of displacement on pressure drag we may assume flat plate valuca and
expect that the error near to the centre line will only have a small ?ffect on
the total drag. In the second example the values are closer together, the
extra thickness only rising to about 5% near the centre line.
6 SOME ACTUAL MAGNITUDES IN A TYPICAL CASE
It may be of interest to give some idea of the actual ma,;nitudes of the
various boundary layer thicknesses near the trailing edge of a fullscale wing.
We consider a delta wing with a root chord of 200 feet, flying at a Mawh number
of 22 at a height of 55,000 feet. 0 is obtained from the first example whilst
8* and S are found on the assumption that the velocity in the boundary layer
follows a 1/7th power law. It has been assumed of course that the boundary
layer is turbulent all over the wing, and that there is zero heat transfer.
Boundary layer thicknesses at the trailing edge
y/s a S' 8
0.05
4111
14511
55.7"
0.2
2.3"
8.2"
31.211
0.5
1.61t
5 51f
21.11'
0.8
0.711
2.611
1'
908
7 THE EFFECT ON TIE PRESSURE DISTRIBUTION
Putting e = I for convenience we may take the momentum thickness o to be
given by
1+1 1
n = U n C (x  s R ( 1
n (1 + 0.1281Jf00
no allowance being made for convergence or divergence of streamlines. and 3
are given in Table I for values n = 4 or n = 5. We shall choose n = 4 as bo ig
slightly simpler numerically, with no loss in accuracy.
Sinoe
8* = He
where H isgiven in Table 1, we have
b~ = 0.0370 12.5 f1 + 0.178M2~  1 x
1+0.128M2
s 71) R7"
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We shall write
81,
and note that for C. = 0.8 (corresponding to n = 1+) MOO = 2,
L = 0.00374.
107we have
The effect of the boundary layer on the flow is the same as though the
fluid were inviscid, but that the wing z = z(x,y) were replaced by
z = z(x,y) + 8'.0
We shall use slender thin wing theory, which is a linear theory. Hence
if uj is the velocity potential due to z(x,y) and a 60 is that due to 8* the
00
two values may be added to obtain the overall velocity potential. 60 is
calculated in Appendix I by methods explained in Ref.5. We aim to determine
Cop, the change in pressure coefficient due to displacement thickness.
The result for Z = 0.8, corresponding to n = !, is, if
11 = Y P
sx
2
= sx0, [K(Iit)  2 log 2{~s
assuming as before that C = 1. The value of K is given in Table 3.
TABLE 3
(12)
Value of K(ri) for Z = 0.8
T1
0.00
0.10 020 030 0.40
C 50
0.60
K(n)
0000
0.004 0023 0.063 0.130
0?233
0.388
0.70
0.75 0.80 085 0.90
0.95
1.00
K(r)
o?627
0798 1.027 1.31+9 1.852
2.8441
00
(is
In the first example, for which s = 1/3, Moo
= 0.577, bop is always positive. Isobars of Aop
2, R = 107, L = 0.00374
are shown
in Fig.4. These
are likely to be reasonably accurate except in the rear part of
to the centre line, where the increasing displacement thickness
an increase in Ao P
the wing near
should cause
It may be noted that we may not suppose that Aop can be obtained from
simple wave theory. We show in Fig5 the value of Aop compared with that
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obtained, by simple wave theory, along the line y/s = 0.225, Similar 4:vergonoos
occur everywhere on the wing.
Aop has a singularity at Ir1E = 1. In fact when r) = 1s it can be shown
that
K(1s) = 43240e02  5.0606 + 0.583c + 0(e 2) ,
and so the singularity is integrable.
8 THE BOUNDARY LAYER PRESSURE DRAG
Once AoP is known the boundary layer pressure drag coefficient is
calculated from the formula
ACD = s r dy r dap ax
Iy/sJ
taking into account both surfaces of the wing. Hence
r1 1
ACD C 4 .l j AoP ax dx ,
o k
or writing k = y/s ,
In our first example we evaluate the integral numerically using equation
(13) near the singularity. We find for R = 107 that
ACD = 0.00008 .
This is only 1% of the invisoid wave drag, which is 0.00821. The wing
considered is rather think (maximum thickness/chord ratio of 11.2 ) and the
invisoid wave drag varies as the square of the thickness, whilst ACD varies as
the thickness. Hence if the maximum thickness of the wing were halved the
invisoid drag would be reduced to one quarter the above value whereas ACS would
be halved. Henoe AC would rise to 2% of the invisoid value. On the other hand
D
ACD. varies a,s R. 2 so that an increase in Reynolds number from 107 to fill snaI
(say 4. x 108) has the effect of halving ACD.
In the seoond example the pressure drag was direotly calculated by the
supersonic area rule. This drag was found to be negative and the reduction in
drag thereby produced amounted to as much as 4? 3; for a Reynolds number of
2 x 106. The results are given in Table 4. With a machine lrogr.e avtilabls.
it was possible to take into account the increased thickonin? of the boundary
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layer near to the oentre line. It was found, however, to make no appreciable
difference to the overall drag. These calculations were performed by
J.A. Beasley, who devised the machine programme.
TABLE 4
Pressure sixag coefficients for the wing; tested by Firmin. M =_22
R 0D Decrease due to
boundary layer
00
000562

107
0.00544
3.2%
6 x 1o6
0.00543
3.4.;r
2 x 10 6
0.00538
4 3,Z
9 THE SKIN FRICTION DRAG
For a thin wing with a boundary layer development as described above the
total skin friction drag will be approximately the same as that over a flat
plate with the same planform. A fair approximation to this may be found by
assuming the plate to be rectangular with a chord equal to the mean chord of
the wing. We may then find the drag in the manner recommended by Monaghan6.
This gives an overall drag coefficient, taking both sides of the plate into
consideration, of
CF
2.8 2.6
= 0.92 TT
104,:.'10 R Ted'
w w
(15)
where R is the Reynolds number based on mean chord. and on free stream conditions
and, in the case of zero heat transfer,
T w = I + 0.1781.1
T
This gives for the wing discussed earlier, with a mean chord of 100 feet,
flying at Mach 2.2 at 55,000 feet
CF = 0.00257
This will apply even if the wing varies in shape and thickness, so long
as the Reynolds number, based on mean chord, is unchanged and the wing is thin
and has a low lift coefficient with attached flow.
If the wing is a delta with rhombic crosssections and Lord V area
distribution and maximum thickness chord ratio 11.20 the wave drag coefficient
is 0.00821, whilst for 5?F; thickness the coefficient is 0.00205. In the
latter case the skin friction drag and wave drag are roughly of the same order
of magnitude, whilst the boundary layer pressure drag is 0.5, of the total
wave drag plus skin friction drag.
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The streamline convergence towards the rear near to the centre line,
ignored in the above estimates, will cause an increased pressure coefficient,
and this, being on backwards facing surfaces, will reduce the Pressuro, drag
slightly in the first example. In the second example the cha,n..c is nc,glig'.c_.t..
10 CONCLUSIONS
The main results are that at moderate Mach numbers:
(1) The boundary layer over a thin delta wing at zero lift develops__rn
much the some way as though the wing were a flat plate of the same pl4nform
placed edge on to the stream, and the skin friction is the some as that of :~.
flat plate.
(2) At test and full scale Reynolds numbers the boundary layer normaz.t
pressure drag is in general small enough to be neglected compared with the
inviscid wave drag and skin friction drag, though this may not be true for
wing with large slopes at the rear, as in our second example.
There seems to be no reason why these conclusions should not apply tc
other planforms besides deltas so long as the wings are slender and thin.
Little experimental evidence for these results is available; however it
was found for the first example that the sum of the calculated inviscid wave
drag of the wing and the skin friction of a flat plate of the same platorm wss
in fair agreement with the measured overall drag the maximum error being about
2%. Agreement was not quite so good in the second example, the error eing
about 5;f..
It is likely that the cause of the disagreement lies mainly in a,rocs 1i
the boundary layer part of the calculations. These ultimately depend on the
assumption of some skin friction law for flow over a flat plate. In view of .he
small effect of pressure gradient which the calculations show, the ute of fla.
plate laws may possibly be justified, but one must remember that Monaghan6 did
not claim better than 10;. accuracy even for flat plate flows.
There are nevertheless other sources of error which should not be
forgotten. One of those is the use of linear theory to determine the inviae:,e
flow. In the first example considered here (Lord v) slender theory lerds to
quite accurate pressure distributions, but it does not do so for the &E:oond
example (Firmin's Wing 3). Calculations by linear thinwing theory gite
improved results for this case, but even so the measured prossurp near to the
trailing edge does not agree too well with calculations. Finally, one must
bear in mind that in the experiments the bands of roughness put on near the
leading edges to induce transition may provide yet another source of error, 1y
spite of efforts made to allow for this.
If boundary layer profiles near to the trailing edges of slender wings
wore measured at a number of spanwise stations it might be possible to obtair,
further verification of the suggestions here presented.
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LIST OF SYMBOLS
alb 1hI,1+IT)I
B see equation (1) and Table 1
C see equation (1) and Table 1
o root chord of wing
0p pressure coefficient
CD drag coefficient
ACD increment in drag coefficient
CF skin friction drag coefficient
D see equation (1) and Table I
E
P1 'P2
H
see equation (1) and Table 1
defined in equations (16) and (17)
I1,Jt defined in equation (19)
k y/s
K(ij) see equation (20) and Table 3
11 index in equation (11)
L coefficient in equation (11) for S
M Mach number
n index in skin friction law Ref.2
P see equation (1) and Table I
r defined by equation (6)
R Reynolds number = /v.
Re ue8/ve
e semispan at trailing edge
S(x) area of section of wing by plane x = constant
s distance measured along streamlines in Section 5
 15 
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LIST OF SYMBOLS (Cont'd)
T temperature.
u,v velocitycomponents in x and y directions
U resultant velocity
V total volume of wing
x,y,z Cartesian coordinates,x along the centre line, the median plane
being z = 0
R M21
Y Euler's constant = 04577216
S boundary layer thickness
displacement thickness
given by 11 = 1e in equation (13)
'q y/sx
6 momentum thickness
o (6/0)1.2
v kinematic viscosity
velocity potential
Euler's c function
Subscripts:
0o refers to values at infinity
e refers to values Just' oiatsid:ethe boundary layer
w refers to values on the su rfaoe of the wing
in refers to values at a "mean" position
 16 
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LIST OF REF RENCES
No.
Author
Tit:Lo, etc.
I
Spence, D.A.
The Growth of compressible turbulent boundary layers on
isothermal and adiabatic walls.
A.R.C. R. & H. 3191. June, 1959.
2
Young, A.U.
Kirkby, S.
The profile drag of biconrca and double wodgc wing
sections at supersonic speeds.
Prococdin,s of a Sympositun on boundary layer effects in
aerodynamics held at the N.F.L. (1955). H.M.S.O.
3
Firmin, M.i.C.P.
Experimental evidence on the drag at zero lift on a series
of slender delta wings at supersonic speeds and the drag
penalty due to distributed. roughness.
R.A.J. Technical Note No. Aero 2871, 1963.
4
Cooke, J.C.
An axially symmetric analo^;uo for gonoral three
dimensional boundary layers.
A.R.C. R. tc I. 3200. June, 1959.
5
V7ebor, J.
Slender delta wings with sharp edges at zero lift.
R.A.N. Technical Note "To. Aero 2508,
ARC 19,5)0. i:iay, 1957.
6
Monaghan, R.J.
A review and assessment of various formulae for turbulent
skin friction in compressible flow.
A.R.C. C.P.11+.2. August, 1952.
7
Erdel i, A.
(Ed.)
Higher transcendental functions. Vol.I.
McGraw Hill Book Co. Inc. New York, 1953.
8
Davies, iI.T.
Tables of higher mathematical functions. V'ol.I.
Frincipia Press, Bloomington, 1934.
 17 
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APPENDIX I
DETERMINATION OF A0 AND pod
The equation of the displacement surface is
Az = 6* = L x
By Roferenoe 5 we have
u
00
F2 =
F1 
A0 = j ~
x F1 + 2 F2
sx
J r aAz x ' log
ax JYY, I t1y'
 sx .
x
AS'(x) log R  fAs"(x) log (xx') dx!
0
AS(x) = 1+
From equation (18) we have
aS(x) _ sx?.F1
?.+1
F2 =
sx
f
0
Az(x,Y) dy ,
, AS' (x) = 14,Lsx4
LS"(x) = 4Ls &x 1
x
41ax log 2p  4Lst f x1 t1 log (xx,) dxr
0
41sx,Tlog 2(3  log x + y + V(,~,k1)
.18 
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Appendix 1
on putts xt = tx in the integral and noting that,, if y is Euler's constant
and *(Q+1) is Eulert s i function7,
1
,~( g+1) .
f t"1 log(1t) at = Y
0
This may be verified by tern.byterm integration and the use of the
series for 4r(e41)7, namely
1
T nTe+n)
n=1
aF2
ax 
1fLstxZ1 log 21P  log x + Y + fir( Z)
(e+ 1 } _ + ,V(~} ?
Now F1 may be written, putting y' = sxt'
ex's (1t') 41 log 1YsxtI + log (y+sxt')] att
j I
1
.eiLsx4 r t4 1 12 log ax + log Ital + log (bt)] dt ,
of
on putting
tt = 1t
Hence
VI
+ t rl ` f t C1
0
19
1 + In 1 , n = y/sx .
_ $Lsx~1 2 log sx + E + Z t'1 [log Ital + log (bt)] at
a = 1IHI ,
T a
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Appendix I
On evaluating by parts of the first integral we may reduce this to
art
= BLsxZ1 IZ + JZ + 2 log 1711 ,
L
1 t~1
t  a at
Cauchy principal values are to be taken where necessary.
Hence we have
A(,p =
2u
u
00
p1 t~1
1~ = f bt dt
0
2 ft5
U00 ax
Lc(r1) 2 log his
K(r1) = IZ  J6  2 log Jr11  2 [Y+ '(&) I .
If g = 4/5 we find from the tables8 that
0.965009 Y = 0.577216
K(r1) = 10.8  10.8  2 log Irll + 0.77559. (20)
10.8 and JO.8 may be evaluated numerically for a series of values Off' T) and
hence K determined. Table 3 gives values of K for a series of values of j.
If r) = 1e, where e is small, K(r)) behaves like a0.2; in fact it can be
shown that for & = 4/5
Y108) = 4?324.0 s0.2  5.0606 + 0.583s + 0(e2)
and so K(ri) has an integrable singularity at r) = 1.
 20 
W.T.59. KS. Printed in England for Stationery Office by R.A.E. Farnborough
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?2 0.000300(,) (FLAT PLATE)
0.1 O.2 0.3 0.4 M/C 0.5 0.6 0.7 0.8 09 10
}+ I VALUES + F ! 1} *2 RY '!`'"  ! HEMS ON AL, A.LC"'LATIVIIS
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FIG.2. CALCULATED EXTERNAL STREAMLINES NO 15.
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2002/1
0082210001000800019
r \
\
;
\ \
1 \
; \
\
r
\
; r
\
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Approves oor*Release 2002/10/16 : CIARDP71B00822R(4301000800019
It
0
It
0
L
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0?007
0?006
0.005
0.004
,~i cp
0.003
0.002
SLEfy ER
oooi
0.2
0.4
0.6
WI G
08 x/c l0
FIGS. ACp WHERE ~/S=0225, BY SIMPLE WAVE
AND SLENDER WING THEORIES.
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533.693.3
A.R.C. C.P. No.696
533.693.3 :
532.526.4
532.526.4 :
TURBULENT BOUNDARY LAYERS ON DELTA WINGS AT ZERO LIFT.
Cooke, J.C. March, 1963.
533.6.011.5
TURBULENT BOUNDARY LAYERS ON DELTA WINGS AT ZERO LIFT.
Cooke, J.C. March, 1963.
533.6.011.5
It is found that, for turbulent flow at Mach number 2 over a thin
delta wing at zero lift, the effect of pressure gradient on the boundary
layer is negligible; thus boundary layer calculations allowing for conver
gence and divergence of streamlines are simplified. When these are dote it
is found that, except near the centre line, where streamline convergence
causes extra thickening towards the trailing edge, the momentum thickness Is
nearly the same as it would be for flow over a flat plate of the same plan
form. This enables the boundary layer pressure drag and the skin friction
drag to be determined simply. It is found that the pressure drag may be
neglected compared with the total drag, whilst the skin friction is the same
as that of a flat plate of the same planform.
533.693.3
532.526.4 .
533.6,011.5
TURBULENT BOUNDARY LAYERS ON DELTA WINGS AT ZERO LIFT.
Cooke, J.C. March, 1963.
It Is found that, for turbulent flow at Mach number 2 over a thin
delta wing at zero lift, the effect of pressure gradient on the boundary
layer is regligible; thus boundary layer calculations allowing for conver
gence and divergence of streamlines are simplified. When these are done it
is found that, except near the centre line, where streamline convergence
causes extra thickening towards the trailing edge, the momentum thickness is
nearly the same as it would be for flow over a flat plate of the same plan
form. This enables the boundary layer pressure drag and the skin friction
1rAg t, ho laraYe?Tnel a1mci+v it is found that ?...h.L^ zSt.r'r c +rag g w~ May b
_.r..,, . . .., ....... .....,. y. E r"
neglected compared with the total drag, whilst the skin friction is the same
as that of a flat plate of the name planform.
It is found that, for turbulent flow at Mach number 2 over a thin
delta wing at zero lift, the effect of pressure gradient on the boundary
layer is negligible; thus boundary layer calculations allowing for conver
gence and divergence of streamlines are simplified. When these are done it
is found that, except near the centre line, where streamline convergence
causes extra thickening towards the trailing edge, the momentum thickness is
nearly the same as It would be for flow over a flat plate of the same plan
form. This enables the boundary layer pressure drag and the skin friction
drag to bt determined simply. It Is found that the pressure drag may be
neglected compared with the total drag, whilst the skin friction is the same
as that of a flat plate of the same planform.
533.693.3 :
532.526.4 :
533.6.011.5
TURBULENT BOUNDARY LAYERS ON DELTA WINGS AT ZERO LIFT.
Cooke, J.C. March, 1963.
It is foi.rd that, for turbulent flow at Mach number 2 over a thin
delta wing at zero lift, the effect of pressure gradient on the boundary
layer is negligible; thus boundary layer calculations allowing for conver
gence and divergwr.ce of streamlines are simplified. When these are done it
Is found that, except near the centre line, where streamline convergence
causes extra thickening towards the traiirg edge, the momentum thickness is
nearly the same as it would be for flow over a flat plate of the same plan
fcrm. This enables the boundary layer pressure drag and the skin friction
drag Lu 'ue :ie.erwlc,ed simpler. i~ is round tnat the pressure drag may be
neglected compared with the total drag, whilst the skin friction is the same
as that of a flat plate of the same planform.
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