ON THE HODOGRAPH TRANSFORMATION FOR
HIGH-SPEED FLOW
II. A FLOW WITH CIRCULATION
By M. J. LIGHTHILL
(Department of Mathematics, The University, Manchester)
[Received 13 December 1947]
SUMMARY
A steady two-dimensional isentropic compressible fluid flow about a contour with
circulation, reducing to the potential flow with circulation about a circle as the
Mach number at infinity tends to zero, is specified in the hodograph plane.
1. Introduction
IN this paper the method of Part I (1) is applied to obtain the high-speed
flow corresponding to the low-speed flow with circulation about a circular
cylinder. The reasons for prtsenting this alternative treatment of the
problem to those of refs. (2) and (3) are again those attaching to any such
multiple treatment: it widens the basis of understanding ?,nd future
development. The work of this part is necessarily more complicated
than that of Part I, however; nor has it the generality and directness in
argument of ref. (2). Its mathematical interest, in the author's belief,
outweighs these objections.
2. The low-speed flow
If the circulation is 477-sina and the velocity at infinity unity, the
complex potential and complex velocity for the flow of 'perfect' fluid
past the unit circle are
, 1 ,o. . ,
. dw . 1 2isinoc
w = z-\ |-2isinalogz,
£ = — = 1 --\ — . (1)
Q/Z
Z
Z
Z
-1
Inverting the latter equation as z = (1—£) (~isina+(cos 2 a—£) 1 ), we
obtain for w the expression
w=
p(—isina-f (cos2a—£)*)+* sin a-f-(cos2a—£)*—
-2isin a log(isina-f (cos 2 a -£) 1 ) = P{Z)+Q(Q, (2)
where
P(0 = isin J l - ^ - l o g a - o ) ,
(3)
and Q(C) can be put into the form
(4)
HODOGRAPH TRANSFORMATION FOR HIGH-SPEED FLOW
443
By expansion of Q in descending powers of (1—4), which are in turn
expanded in ascending powers of £, and by inversion of this double sum,
we find that the branch of Q(t,) equal to 2(cos<x+asina) when £ = 0 has
for its Taylor series near £ = 0 the expression
n=0
'
\
?>•
The hypergeometric function appearing here is an integral function of n:
it has the integral representation (for all n)
sln'a
(cosa)1"2"—(n—£)sina: J aH(l— x)-*-ndx,
(6)
o
which shows that it is 0(|w(cos2a)-"|) as \n\ ->oo for 9?(») > 0.
Hence the series (5) converges for |£| < cos2a. When 5R(n) < 0 expression (6) becomes
I
— (n—£)sin a J x~*( 1 — x)-*-n dx+ O(|7i(cos2a)-n|)
cos2a)-"l) = 0(|n|). (7)
The series for Q(X) can be continued in the manner adopted in § 2 of
Part I. In fact the series is
j(-iH-^im--»' J . ( v _ i _ i ; i ; 8 i n M ( _ e r ^
,„
-ooi
within its circle of convergence |£| < cos2a, being minus the sum of the
residues of the integrand at its poles v = n ^ 0 to the right of the contour;
while for |£| > 1 the integral, by (7), is equal to the sum of the residues
of the integrand at its poles to the left of the contour, i.e. to
2
Here, as in Part I, if arg£ > 0, arg(—t) must be taken as arg£—n in
order to secure the convergence of the integral (8), so that (—£)l~" is
—i(—l)n£*-n; while if arg£ < 0 then arg(—£) is arg£-f-7r and (—£)*"" is
i(—l)np-n. Thus, for |£| > 1, if ^ signifies the sign of arg£ on the path
of continuation,
n=0
444
M. J. LIGHTHILL
This can be continued into the region cos2a < |£| < 1 by use of (7), whence
0,0 _
T
n=0
(11)
plus a function analytic for |£| > cos2a; (11) in turn differs by another
function analytic for |£| > cos2a from
=F»sina[(l-{)-i+log(l-0],
(12)
with which equation (3) should be compared. It is deduced that
P(£)+(?(£), continued analytically via the region arg£ < 0, has no singularity at £ = 1; but that if continued via the region arg £ > 0, it has a
singularity there, that of — 2isina[(l —£)-1-flog(l—£)]. Also, by (10),
<2(£) changes sign in going round a cut joining cos2a and 1. The Riemann
surface corresponding to that in §2 of Part I can now be depicted as
below, Fig. 1. The values of w shown are those obtained by continuation
an? 5=0
FIG.
1.
along paths not crossing a cut from infinity to the singularity 8: w is
one-valued on the surface so obtained. AB is the cut joining the two
sheets. A positive encirclement of S adds <tn sin a to w. The only singularities of w are S and B, corresponding in the physical plane to the point
at infinity and a certain finite point C where z = — i cosec a. The cut AB
corresponds to a circle of diameter 0C (O the origin) which passes through
the two stagnation points (corresponding to A): on this circle 6 = 0.
3. The high-speed flow with fjr^) = e-"*»
To obtain from this a high-speed flow, we write, as in Part I, § 3, corresponding to the above P, Q,
P = - i s i n a f (l-n-1)^n(T)e-«c>+f9),
(13)
n=l
Q=
(n
2 »w-^ )! F[n~h ~
n =o
o
n
^
2
''
HODOGRAPH TRANSFORMATION FOR HIGH-SPEED FLOW
445
and consider 3(P-f-<2) as a commencing branch of 0. The former series
converges for T < TX, the latter (by equation (6) and the asymptotic
formula for ifin(r)) for e8 < e"1 cos20: since .1 is an increasing function of T,
this can be written T < T2 for some T2.
When T <T2we have as in (8)
-0"*
(15)
with the ^F sign according as 0 $ 0. For T > r1 this is equal to the sum
of the residues to the left of the contour, i.e.
Q =T
n=0
4*
*
"
(,—hV-
The latter series, which we call R, is convergent for all r, Q. Therefore
the continuation of Q round a cut from T = rlt 6 = 0 to r = T2, 6 — 0
is 2i?—Q; and continuation a second time round gives again Q.
Also, continuing the series for ^(Q—R) in (16) into the region
T2 < T < TV we obtain
= » V i±IW«0|_n(T)e<*-«*+<«>
(17)
plus a function analytic for T > T2. Expression (17), for T > TX, is equal
(since the residue of n/coaiTv at v = \—n is (—I)™-1) to
r
°°^
I
^
sina f . J ^ ^ ^ j ^ ^ ^ ^ ^ ^ ^
I 2ni J cos 77v v
J
( l g )
according as d $ 0, where
X = sina f 7r(w+l)Cni/rn(T)e"<8»+f5)
(19)
n=0
n=0
(with Co = 1, C1 = 0 as usual, the term n = 0 coming from the pole at
v = 0). X converges everywhere. Hence forr < ^expression (17) becomes
—isina V ( _ l ) » - i , ! l | ^ , ( T ) ( _ i ) » c ^ + l ) | . , t i ^ j ;
5092-4
(20)
446
M. J. LIGHTHILL
and so when +~(Q—R) encircles T = rx in the positive direction (considering (T, — 0) as a right-handed system of axes, so that 0 < 0 comes
first in an encirclement of (T1, 0) in the positive sense starting from T > TX)
it increases by — 2X. Hence Q 'has a period of + 2X at r = rx' (i.e.
increases by this quantity when the point is encircled in the positive
sense), according as it has been continued from its starting-point via
0$O.
We also have, for T < rlt 6 $ 0, with the positive integers the only
poles to the right of the contour,
aoi
P = t s i n a - L f _^_/i_iU( T )e'<«''-8.-i9)dv.
2m J 8in7rv\
v)Yv
(21)
i
When T > Ty this is the sum of the residues at the poles to the left of the
contour; these are all double. Hence,
00
P = isina y
i
(— 1)" —
n=0
= i sin aJ J J |Wn)[l
n =0
- -)
-ftsina J
(=F»ir)(l + i)(-nOw^n(T))e'*'+*«»,
n=0
*
(22)
'
and the second term is ^X. Hence, when P encircles T = TX in the
positive direction starting from T < TX (SO that 6 > 0 comes first) it
increases by 2X.
Thus, if continued via 6 > 0 (right-hand sheet), P + Q has zero period
at T = Tj, 0 = 0: in fact there is no singularity there at all since, by (21)
and (18), P+Q differs by a function regular at r = rv 6 = 0 from
2-ni JJ LLCOSTTV
i
sinTrvJ v
YvK
'
v
'
_ooi
which converges uniformly in the neighbourhood of (TX, 0) since the term
in square brackets is 2me~i7"' cosec 2m>. Thus there is no singularity of
P+Q (except B) on the right-hand sheet: on the left-hand sheet, however, there is a point S where P+ Q has period 4X. If a cut is made from
this point to infinity P+Q is one-valued on the cut Riemann surface,
taking values as shown in Fig. 2, in the different regions. As ip = 3 ( P + Q)
encircles the point S (which corresponds of course to the point at infinity
in the physical plane) it increases by 3(4X). By (19) this is not zero for
HODOGRAPH TRANSFORMATION FOR HIGH-SPEED FLOW
447
all T, 6: hence the stream-function is many-valued in the physical plane
and the method (when /n(Tx) is taken as e-"*1) is useless in this example.
This is due to the presence of circulation.
FIG.
2.
4. The correct value of /n(TX)
To overcome this difficulty I thought of choosing a different fn(r^) more
symmetrical to ifin(r). The reasons for this are far clearer from. the
method of ref. (2) which I evolved subsequently. ^_n(r1) was first tried:
this got rid of part of the period 3(4X) and pointed clearly the way to the
correct value, whose raison d'etre is demonstrated far more generally as
well as convincingly in ref. 2,
£=»ilL>,
(24)
where ifi'_n denotes the derivative oiifj_n. The work is here displayed with
this latter value used.
Consider, for 6 $ 0, the integral
on
- f
"7rt
J
-a*
(-*)!
X
(25)
l—i.
"Eor T < T2, this is equal to the sum of the residues of the integrand at its
poles on the right (these are double poles at the non-negative integers), i.e. to
|
x
1—n
(26)
448
M. J. LIGHTHILL
which we write as Q±T, where T converges everywhere and T-> 0 as
the Mach number at infinity tends to zero. Also, as this happens, the
integral (25) tends to the low-speed integral (8), since if>v(r) ~ T1",
: =
1—v
—=- ~ -^
:
=—
1—v
= rx "'•,
(27)
and (T/T^'e-'" 9 = [(g2/go)/(1/5o)]i"e~<"9 = (fle-1*)" = £". But (8) is the
low-speed Q. Hence the Q here defined tends to the low-speed Q as the
Mach number at infinity tends to zero and so is a reasonable extension
thereof.
When T > rlt the integral (25) is equal to the sum of its residues at
\—TO (TO ^ 0) and —n (TO ^ 2), i.e. to
«
/qivw
n-o
n
i\tn
'*
*''
n-2
3\|
^
fl)
which we write as ^W-\-R, where R converges everywhere. Thus the
continuation of Q for r > T2 via 0 $ 0 is ^ ^ ( r + T f ) , and its continuation right round a cut from T = TI? 5 = 0 to T = T2, ^ = 0 is 2R—Q; and
continuation a second time round gives again Q.
Now W can be rewritten as
^
n-0
plus a function analytic for T > T2. This series is
r
"^
i.
27n
_ci
COS77V
"
according as 6 $ 0, where
CO
n=0
»
)
«
(29)
HODOGRAPH TRANSFORMATION FOR HIGH-SPEED FLOW
449
which converges everywhere. For T < TV therefore, (29) becomes
±sinaf |
L
(n+i)-V» + iW(^- n -j(T 1 )+2r 1 f_ n _ l (T 1 ))(Ti)e^ n+iWfl -
n=l
- 2 ^(T)Cn(i(T 1 )+2r 1 ^(r 1 ))e-H=FI = ZTY, say,
Y = naina f Cn4ln(r)(,!,n(T1)+2T1^n(r1))(e^B+e-ine),
where
(32)
J
n=0
(33)
n=0
which is purely real. Thus W, when continued round r = T1; 6 = 0 in
the positive direction (6 < 0 first), decreases by 27: hence the period of
Q = R^^T+W)
is ± 2 F according as it has been continued from its
starting-point via 6 £ 0.
Consider next the integral
isina-L f -
(l
sin TIT \
27rt JJ si
(T)
vj
(34)
1—
—avii
which when r <r1ia equal to
-isin. J
*Uv^
-isina J (Tfa)fl--k(T)i^> ( T l ) + a r ^ ( T t ) e ^ = P±{Y-X),
(35)
with X, Y as in (31), (33). The P here denned reduces to the low-speed P
as the Mach number at infinity tends to zero, by arguments similar to
those for Q. When T > TX the integral (34) is
z
n-0
+tsin« ^ ( ^ H ^ W ) ^
n-0
1
^ ^
1
^
= HTX,
say
*
(36)
Hence P continues for T > TX into H^fY, so that in encircling T = Tj,
5 = 0 in the positive direction (6 > 0 first) it increases by 2y. As in § 3,
therefore, P+Q is one-valued on the Riemann surface cut to prevent
encirclement of S, is regular except at S and B, and has period 47 at S.
\\i, however, being the imaginary part of P-f Q, has no period at S since
Y is real, and so is one-valued on the uncut Riemann surface. The
solution for tfi is therefore quite possible physically; and an exactflowof
450
HODOGRAPH TRANSFORMATION FOR HIGH-SPEED FLOW
a compressible fluid past a contour with circulation has been mathematically specified.
The series for ifi in different parts of the Riemann surface are indicated
in Fig. 3. The value when T2 < T < TX is obtainable by splitting W as
FIG.
3.
in (29) into a term continuable round T = TX and one convergent for
T > T2, and using ^ =
^(P+B^T^W).
Strictly, an investigation into the one-valuedness of x, y.as well as that
of ifi is necessary, since x, y may have constant periods though ifi has none.
Further work is omitted here, however, since the required investigation
is given in ref. (2). It is clear that the method given there is greatly
preferable to the present one, as well for the reasons given in § 1 as for
the fact that it gives a single series for ^ convergent throughout r < rg,
which will be more useful than those of the present paper especially around
T = TX and T2 (series of the type of (47) of Part I will also be convenient
in this region): this single series is also such that it does not involve
relations in the low-speed hodograph plane, which our §2 shows to be
rather involved in even a simple example.
REFERENCES
M. J. LIGHTHIIX, and J. W. CBAOGS, 'On the hodograph transformation for high-speed flow. I. A flow without circulation': see above,
pp. 344-57.
2. M. J. LIGHTHIIX,, 'The hodograph transformation in trans-sonic flow. III. Flow
round a body': Proc. Roy. Soc. A, 191 (1947), 352-69.
3. T. M. CHERRY, 'Flow of a compressible fluid about a cylinder. II. Flow with
circulation': unpublished sequel to Proc. Roy. Soc. A, 192 (1947), 45-79.
1. S.
GOLDSTEIN,