TECHNISCHE HOGESCHOO^ VLIEGTUIGBOUWKUNDE
rmm:--'-^ - -~ -,^^tict 12 Juli 1950
REPORT NO.9 Kluyverweg 1 - 2629 HS DEI F T " O c t o b e r , 1 9 4 7 . T H E C O L L E G E O F A E R O N A U T I C S G R A N F I E L DOn source and vortex distributions in
the linearised theory of steady supersonic flow
by
-A. Robinson, H . S c , A»P»R.Ae.S. oCo
.SUMI4ARY
The hyperbolic character of the differential equation satisfied by the velocity potential in linearised supersonic flow entails the presence of fractional
infinities in the fundamental solutions of the equation, Difficulties arising from this faot can be overcome by the introduction of Hadamard's "finite part of an
infinite integral". Together with the definition of certain coun.terparts of the familiar vector operators this leads to a natural development of the analogy between incon^jressible flow and linearised supersonic
flow. In particular, formulae are derived for the field of flow due to an arbitrary distribution of supersonic soua?ces and vortices.
Applications to Aerofoil theory, including e calculation of the downwash in the wake of an
1. Introduction
It is well known that the elementary solution of Laplace's equation in three dimensions - i.e., the velocity potential of a source in Hydrodynamics, and the potential of a gravitating particle in Newtonian potential theory - has a counterpart in the linearised theory of supersonic flow, viz., the velocity potential of the so-called 'supersonic source'. However, the development of the analogy meets with obstacles which are largely due to the fact that the velocity potential of a supersonic source becomes infinite not only at the actual origin of the source but also everjwhere on the Mach cone emanating from it. Thus, in trying to
evaluate the flow across a surface surrounding a
supersonic source, the resultant integral becomes infinite, This and other difficulties can be overcome by the
introduction of the concept of the finite part of an infinite integral, which was first defined by Hadamard
(ref.l) in connection with the solution of initial value problems for hyperbolic partial differential equations. A description of this concept' is given in para.2 below, and applications to problems concerning source and doublet distributions under steady supersonic conditions will be found in paras. 3 aJ^d 4.
The concept of a vortex ia the linearised theory of supersonic flow was first considered by Schliehbing
(ref.2) who obtained the field of flow corresponding to a 'horseshoe vortex' by a synthesis of doublets.
Schlichting's approach has been the subject of some
criticism as in certain respects the supersonic horseshoe-vortex is different out of all recognition from its
subsonic counterpart. However, it is shown in para.5 below that more generally the field of flow due to an arbitrary vorticity distribution, \mder steady supersonic conditions can be calculated in strict analogy with the
method due to Stokes and Helmholtz in classical Hydrodynamics. The results are in agreement with Schlichting's for the
particular case of a horseshoe vortex.
Applications to Aerofoil theory will be given in a separate report.
2. The Finite Part of an Infinite Integral
Let D (x,y,z) be an algebraic function of three variables so that the equation D(x,y,z) = 0 determines a surface 21 iii three-dimensional space.
The surface S divides space into disconnected components 2j2 in which D(x,y,z) is of constant sign; also, D(x,y,z) will be supposed to change sign across any orviinary point of 21 • Further let f(x,y,z) be a real function defined in a certain region R such that
+ S^(x,y,z)D" -^g^^^(x,y,z)D''t^...^gj^^^(x,y,z)D''""*, (1) /Where,..,
-3-where n is a positive odd integer, n = 2 k-t-1, k = 0,1,2,,.,
and the functions g(x,y,z),
ërji^tjfz),
,.. are either all
analytic everywhere except on 21 » or at least have derivatives
of a sufficiently high order, which are bounded in the
neighbourhood of 21. At any rate it is assumed that the
analytic expressions for these functions may be different
for the different S .
n
Given a small positive quantity €
,
we denote .
by N(€) the set of all points s which satisfy an
inequality js - Sqt:$£ for at least One point Sp of 27 .
We denote the boundary of N(e) by B(€-)» a^^ ^e Senote
by R(€ ) the region obtained by excluding from R all the
points of N(€). Furthennore, given a curve G, a surface
S, or a volume V In R, we denote by C(€), S(€), and
V(€) the subsets of G, S, and V respectively which are
obtained by the exclusion of the points of N(€).
The concept of the finite part *J of an
(finite cr infinite) integral J - where J is any
line, surface, or volume integral of f on C, S, or V
respectively, e.g.
fdxdy, j fdxdydz,
e.g..f
fclx,
I
Jc
-'S
where C, S, and V are supposed to be bounded - will
now be defined as follows.
Given the formal expression for J, we denote
by J(€) the corresponding integral taken over C(e),
S(e), or V(e) only. Subject to the specified conditions
of regularity, J(e) will be finite and of the form
J ( 0 = a ^ e
f.,,. + a^
,e +0(1),..(2)
where 0(1), as usual, denotes a function which remains
finite as e tends to 0. We then define * J by
-n/2tl
.^\
= lim. /J(€) - a £ - ... - a e J
€-^0V • ° . k-1 / ...
J
As stated in the introduction, the concept of the
finite part of an infinite integral is due to Hadamard
(ref.l), whose definition, however, applies to a more
—-restricted type of integral only, Hadamard writes /j
instead of our *J which is used by Courant and Hilbert
(ref,3).
It will be seen that if J is finite then J = j,,
Also, the finite part of an (infinite) integral is
invarient with respect to a transformation of coordinates,
provided the Jacobian of the transformation does not vanish
on S . In particular, if we are dealing with the finite
parts of integrals involving vector quantities, the result
is independent of a rotation' of coordinates.
/There....,
(3}
There will be no occasion for confusion if in future we refer to the finite part of a (finite or
infinite) integral simply as 'a finite part', The finite parts of m-fold integrals in n-dimensional space, n> 3» m < n can be defined in a
strictly analogous manner. The rules valid for them are,mutatis mutandis, the same as for finite parts in three dimensions.
The rules of calculation with finite parts, such as the rules of addition, are the same as for
ordinary integrals. Also, iff depends on a parameter A , but D is fixed, • then - provided the g f-unctions are
STAfficiently regular (e.g., if they are analytic in the various ^ - it is not difficult to show that we may differentiate londer the sign of the integraal, e.g.,
d_
dX
ff^1=ït^^ '^'
Under similar conditions the finite part of
a multiple integral may be obtained by successive integration (including the operation of taking the finite part) with
respect to the independent variables involved, taken in any arbitrary order. Thus, with the appropriate limits v/e have,, for instance,
7 fdxdydz =
1(^
[( ƒ ^^x) ^Y) ^^
(5)
More generally, we shall encoimter cases whereD, and therefore 21 , depends algebraically on one or more parameters. We are going to show (i) that even in
tha.t case we may 'differentiate under the integral sign' and (ii) that if a given integral, or finite part, involves integration with respect to such parameters, as well as with respect to one or more of the space coordinates, we may
exchange the order of integration without affecting the value of the integral.
To see this, we increase the ordin.ary three
dimensions of space x, y, z, by the parameter or parameters involved. Then in the augmented space, the surface D = 0 is again fixed, and in order to prove our assertions, it is
sufficient to show (i)' that in order to find the derivative of a finite part in n-dimensional space with respect to
any variable which is not involved in the integration, we may differentiate under the sign of the integral, and
(ii)'that for any multiple integral in n-dimensional space, 1 < m ;$ n, we have
3t^ , x„ - *r
r / ftdx-j^ dx ... dx = j fdx^ dx ...
dx m taken over the appropriate regions. It is clear that (ii)' would prove, (ii), by industion •-5-We may reduce (i)'to (ii)I In fact, (i)'
states explicitly that
m mi
and this will be proved if it can be shown that
r f
dx ,.. dx=
fff
^ ^ dx-i
... dx /^ dx -t C,J 1
m-1J{^J
- ^1 ni-i;
HIwhere the lower limit of the integral with respect to x
is arbitrary and C is independent of x^^.
^
Putting 2) f
p = ,
we have
f = f -I- fpdx ,
•J
o / m
where f is the value of f for an arbitrary but definite
value of*^ Xjjj [for given x ,,.., x ^^ , and the integral is
taken with that particular vali;ie of x^, as lower limit.
Now, assuming that (ii)'has been proved we have
J ' ( J F d x ^ ^ dx^ ... dx^^^ = J ( ^ J p d x ^ ... dx^_]^dx^
and| so
*''f - f;) dx^... ,,^_. J ( ^ j r | ^ ax^... dx^^^dx^,
0 i.e. fJ
f dx . . . dx T =^r/ ^fÜJ. dx^ , , , dx ^ dx-jTf ^dx ^«.
1 m-1 j (^ j Oxjj^ 1 m-y nTJ ° ^
dx ,
and the last term is independent of x , as required.
To establish (ii)5, we have to prove
{( ft dxNdx , . . dx , = I f dx . . . dx . . . ( 6 )
J \ ] m / l '^^ J 1 ^
Putting I f dx = F, we see that (6) becomes
<J m
"'^ "^ ^ dx, . . . d x ^ (7)
* r * r c) F
F dx^ • # • dx^ T = I :=r— dx,
taken over a certain region R on the right hand side
and over its boundary S on the left hand side
respectively. This is essentially the theorem of
Gauss (cr Green) for higher spaces. For m ^ 3,
this theorem will be proved below for finite parts
(without relying on the results of the present
discussion), and the proof for greater m is quite
similar.
An important example of a finite part vd.ll now 2 a
be calculated. Let D(x,y,z) be defined by D 2 x2
^f^^^y^^T)
and f (x,y,z) by
' f ( x , y , z ) = p o f^2^ ^^.-j V^ fcr x2>/32Cy2t4 ^ > 0
and
_f(x,y,z) = 0 (8)
elsewhere where CT and /J are arbitrary constants. We
find that all the oonditions laid down at tfee beginning
of this paragraph are satisfied in every region R not
including the origin, the surface 2J being given by the
cone x2
-0^
Cy +•27= 0.
Further, let the open surface S .be givf
=0^1 y + ss ,^ r^ where o( > 0 and r >-^ . S
be given by
i s a
c i r c u l a r area including the 'circle x =0^ , y^ -j- zr = - ^
/32
'
on which f beoomes infinite of order 3/2. We are
going to evaluate *J =*|^ f dydz
Given € > 0, let S (£) be the points of S(e)
for which
^^>Q^^o'^ ^J\
and
Zni^)
'the complementary
set of S(€). Then f vanishes on S2(€) and so
J ( e ) = / f djid.;}^ = f f dMd^ ^ I f d M ^ = / f dadat =
r <r oc dud^ _ c r ' (^ fidp 4Q
where
y = ^ ^ cos 9, z = ;5- P sin 0, and "^ is the
_ c<radius of the circle bounding S (€). It is easy to
deduce from the definition of S(€) that (compare Fig.l)
We then obtain
/3
1
-7-and so
cr X dy dz 2'ïr<y (Q)
This result is of fundamental importance for siibsequent developments.
We shall also require extensions of the divergence xheorem of Gauss and Green and of the curl theorem of
Stokes to finite parts. Particular cases of the divergence theorem are in fact proved and applied by Hadamard in the above mentioned book.
A function f will be called an admissible function if it satisfies the conditions laid down at the beginning of this paragraph, A vector function f will be said to be admissible if its components are admissible - and this is independent of the system of coordinates. If all the first derivatives of the components of f are admissible, then div f also is admissible. We are going to show that imder these conditions we have for any volume V bounded by a s^urface S such as considered in the ordinary divergence theorem,
div f dV .(9) By equation (1), the vector f = (^f . f , f ) . n p -i^
can be divided into two parts, P =CF^, F 2 , F 3 j,and G 4 G jG-fjö /? f, = "P4'^,- so that the • 5^ are finite and continuous onZ" while the G^ become infinite there. Then div F is either finite
and continuous everywhere in its domain of definitions or it becomes infinite of order -g- on X.. Even then f div F dV exists as an ordinary improper integral, and jg FdSr^s
f div F dV. Since div f = div F4-div G, it is "Tiheref ore Jy-- - - - -^
sufficient to show that
^s ~ Jv
div G dV
i.e. f
ƒ G ds - ƒ div G dV = 0 .-(10)'
Let S.'(e.) be the product of (the set of points common to) V and B(€). Then V(€) is bounded by
S(C)i-S'(€), the sumset of S(e) and S'(e). Hence, applying the divergence theorem to the volume V (€), we obtain
/ G dSt ƒ G dS = r div G dV
•^sier
" Js'(£) " ~ Jv(e)
ƒ G dS - / div G dV = - / G dS ...(11)
-^s(e) "
~^V(€) " Js'(€) " "
Now / G dS is of the form
-(n/2 - i)
where H(£ ) is a function which tends to 0 as C tends to 0,
(€)
and similarly f div G ds will be seen to be of the form
i
- . n / 2 ^ ^
div' G dV = b € ~" ^ + ... + B, £ " i-/ div G dV-V K(6)
V(e) "• o k J y
-where K(e) . is a f-unction which tends to 0 as C tends to 0.
In other words ƒ G dS and f div G dV differ
^ s Js(e) -^v(€)
from / ;'G dS , and f div G dV respectively only by vanishing
functions of C and by fractional infinities of €. . Hence,
in order to prove (10), it is, by (11), siAfficient to show
that
L
-
n/2 -* , ,
G dS = c € ^,,,4,c^£ ^L(e)..(12)
(e) ~ o k
where L(€) is a function which tends to 0 as €. tends to 0.
And (12) can be readily deduced from the fact that the
components of P satisfy conditions of the type indicated
by (1). In fact (1) shows that on S'(e.) G 1 is of the type
G ' ^ = C Q €
-v.,.,+ c^e (13)
where the C depend toi the parameters of S'(G)» and similar
expressions hold for the other components of G.
Next, let f be a vector fi:uiction of the same
description- as before, and let J be an open surface
bounded by a curve 0 such as considered in the ordinary
curl theorem. Under these conditions we are going,, to
show that
(14)
ƒ f d^ =
(
curl f dS
Jc "" Js
Splitting f into two parts F and G as lief ore,
we first show that"" ""
c u r l G dS (15 ) .
TGÖI = r
^ C "^S
-9-Let •C'(€) be the product _set of S and B ( 0 , Then S(e) is bounded by C(é.) -f- C'Xfe) and so, applying Stokes' theorem to S(e), we obtain
/ G d ^ . j GdJ? = / curl G dS ....(16),
^G(e) ^*'G'(e) Js(e)
In order to be able to deduce (15) from (16) we have to show, similarly as in the proof of the divergence -theorem that
-n/2 -i
G dR = c £ ^...+ c€. -fH(€)..(17) 'C'(£) - " 0 ^ k
X
where lim L(6) = 0, and t h i s follows from ( 1 ) , a s Tiefore.
6-^0
We s t i l l have to prove t h a t
F'd£ = I curl F dS (18)
ƒ
This is obvious, by Stokes' Theorem, if curl F remains finite everywhere, and if curl P becomes infinite ora S (in which case the right hand side^of (18) is an ordinary improper integral), provided S has not got a finite area in common with 21 • Assuming on the contj;ary that S has a two dimensional subset S bounded by "C in common with ^ , it is then sufficient to show that
j £ d « . r
«-'n J S
curl F dS (19)
Again, since curl F is an admissible vector, it follows that F is of the form P = F-j_4- IV, where the
components of F are finite and "continuous on 2 1 an^ the components ^ of F vanish on %. (BO tha1i F \^ "^
remains bounded as €: "^ tends to 0. ) -"^ Hence
f F dj^ == f F., dJ?
J c " ~ Jc""-^ "
On the other hand by the definition of the finite part,
T-
Xr
curl F dS = curl F dS, and by Stokes' theorem
s JS "^ ""
f o r f i n i t e functions ' f F dJ? ^ I c u r l F dS
J c ~L "• - J s 1 "
and so / Z ^ = | curl F dS, as required.
J'c -''s
Equations (19) is now established completely.
3. First applications to the linearised theory of steady
supersonic flo'w
In this and the following section we are going
to discuss- solutions of the equation
.ft2 -i.i4^2£i| + ^ . 0 (20)
'^ 'B x*^ -c:' y"^ c) z^
in relation to the linearised theory of supersonic flow.
The following details are not intended as an exhaustive
introduction to that theoryj their purpose merely is to
establish and explain the terminology used in the sequel,
Assume that the free stream velocity of the
given field of flow is parallel to the positive direction
of the X-axis ahd is of magnitude U , where U is greater
than the speed of sound ^
J^
,
Calling the total velocity components in the direction of
the Xj y, and z-axes, , u, Vj and w respectively, and
assuming that u is large compared with v and w,
and compared with its difference from U, we obtain, for
steady conditions, the linearised Eulerian equations
- 1 '^P =u' ' 9 u
^ x
- ^ p - ^ y - ^ P^ x
= u "^ V
'o' X = TT ^ W- 1
Q y ^ x
_ (21)
c? z
'^
where the terms of second order magnitude have been
neglected.
Under the same assumptions, the equation of
continuity which, in full, is
2u_^Qv;2Z^_J: /^Ji li+JL ^ + J1. l i V 0 , ..(22)
' ^ x ^ y ' c l z a2
K, P "è :L P
'c)y
f> ^ z I
-11-becomes, taking into account (21),
n 2 l u ^ ' l v ^ ' ^ J l = 0 (23)
^ x Q y ^ z
2
whereƒ3 = M - 1 = - 1, M = -2— being the Mach number.
a^ a
Equation (21) is the linearised equation of continuity. If, in addition, the flow is irrotational, then we have curl 3.= 0, where 2," = (^> "^t w ) , i.e.,
^ i j l ^ ^ = 0 • " ^ J i ^ ^ = 0 "c^ V / ^ ^ - n..(24) 3 y «5 z '^z c)x "^x '^y
In that case, there exists a velocity potential
so t ^ t 3, = - grad ^ , Expressing u, v, w in terms of 0^ in (23), we obtain (20).
Equation (22) expresses the fact that div q' = 0 where the 'current vector* g;' is defined by q'= /^ q_. Since (23), which is the linearised version or (22),
indicates that dlvQ'C -fS ^1 v, w'j] = 0, it will be seen I that the corresponding 'linearised current vector' is '^'=
pQ-/S^i V, w ) , where /o is. now constant.^ Dividing g.* by/o we obtain a vector ^ * = . (^ H ' S ^ U , V , W/ which will
be called the reducea current velocity, or short c-velocity of the flow. Thus, apart from the flow of q, across a surface S, J q dS we are led to consider als'o the flow of 2^' across ö.~ In order to distinguish between the
- two types of flow, the flow of c^' will be called 'C - flow', It will be seen that the linearised eqxiation of continuity
(23) is the differentiaal expression of the fact that the total c-flow across a closed surface vanishes.
It now becomes natiAral to introduce alongside the conventional operators V , grad, div, and A , the opera+-ors
V|i/9, gradh/3, divh/3,Ali^. (Read 'hyperbolic nabla of index/?', 'Hyperbolic gradient of index^, etc. The index/3 will normally be fixed throughout and may therefore be jomitted,^ -The operators Vli-/2 will be defined by
V ii/3 =(-^2 _ ^ ^ ^ ^ ^-^ h and gradh/O and divh/?,
^ ox '^y ^ z-^ '
as the two modes of this operator which apply to scalars, and to vectors in scalar multiplication respectively. The operator A h o is then defined by Ahfl= div gradh/3 = divh^3 grad. Equation (20) may now be written^ h i = div gradh|_ = divh grad % = 0 (25) By the divergence theorem we laave, for any fimction
which is sufficiently regular on and inside a closed s\Arface S bounding a volume V,
j gradhi dS = f A h f dV (26). J s Jy
If, in addition, ^ ^^ a solution of (25) then
gradh J dS = 0 (27)
i'
However, if we replace ordinary integrals by
finite parts, then equation 1.26) holds even when infinites
are involved provided gradh x and A h ^ are admissible
functions with respect to a certain surface "2. « Thus,
in that case
x_
^
gradh5clS = / A h $ , dV (28)
T
gradh 5clS =
r
-/S J V
while (27), becomes
gradh $ d S = 0 (29)
.In the
composed of cones
of the form '(^-x^ ^ - ^ ^ py-y^y+fz-zJin . o
sequel, such surfa-jses IE yill be frequently
We notice that if a function "^ is admissible
with respect to a surface 51 and another function
Ó
is
regular (or has derivatives of sufficiently high order)
on X , then the product
"ir ^ ±s
admissible, (The
product of two admissible funotions is not in general
admissible). We also notice that if a finite number of
functions are involved, admissible with respect to different
surfaces2^^'' , then they will all be admissible with respect
to one and the same suiiac^, viz,,' the sum of the different
X '^»/S^ven by the product of the funotions D^"-^ defining
the X. ^^^.
Let ^ l°e a function which is regular (or has
derivatives of a sufficiently high order) on and inside a
volume y bounded by a surface S, and 'Y' a function which,
together with its first and second derivatives is admissible
in y (with respect to some algebraic surface
%,
). Applying
(9) to f =T|r gradh 5" , we obtain
^gradhfdS =
j
grady-gradh$ dy+
f i^^hi
dy .(30)
J V J V •
and similarly
f ^ gradliY^S. = I grad $ gradh' 'ijr dy-»- j J A h § dy
J s -^V ^V
• . - . . . ( 3 1 )
1 3
-N»
w grad y gradh I . - / S ' M
1ÈJ,2Ï
M ^ l l l l
^T^ ^% ^'^ b V ^"2. ^ 2 .= gradh Vf grad $ •
Hence, subtractingf (31) from (30;, we obtain
X
y gradh I. - | gradh y") dS = frf AKI"-#AKljf)dy ..
. . . . ( 3 2 )
This is the counterpart of Green's formula,
extended however to include finite parts; (compare refs. 1
and 3 ) .
4, Source and doublet distributions in steady
supersonic flow
Elementary solutions of equations (20) or (25)
are the functions
w
(x, y, z) defined by
$^(x,y,z) =
and
J (x,y,z) = 0 elsewhere
where P = ^XQ , y^, zJN and CT are arbitrary. ^ p will
be said to \ie the velocity potential of a source of
strength
a:
located (or, 'with origin') at P.
The actual velocity potential of a (weak) source travelling
at a velocity -U in a field of reference travelling with
the source, is olitained by adding -Ux- to
$
as given
l»y (33). For reasons of eimplioity ,f p '^
as in (33) will lie called a source for positive as well
as for negative Of .
Similarly, a function Ilj-p will T»e said to be
the velocity potential of a counter-source of strength (T
located at P j if it is given by
llr (x,y,z) = • ^ ===- for
(x-x-^y/f^
x<^x
0and
IK, (x,y,z) = 0 elsewhere
/
34)
A 'doublet' is obtained by differentiating ^ with respect to length in any given direction, the P differentiation being performed relative to the coordinates cf P. Thus the velocity potential 9p, of a doublet whose 'axis' is
parallel to the z-axis is given by
J (x,y,z) = p
1
^ ^ = \
M'-«'[Cy-0'+(-O']]
3/2
\for
C ' M >/3 K^-^c^) t {---^
and x:)x o fand
/.(35)
f p(x,y,z) = 0
elsewhereA 'counter-doublet' is obtained by applying a 'similar operation to "U''p. An asterisk will be employed to indicate fundamental solutions of unit strength (e.g., $ * ).
It will be seen that the velocity potentials of sources, counter-sources, doulflets, etc., and all their derivatives are admissible functions in all regions excluding their
origins, the surface ^ being given by
.x^-^2 _ ^ 2 ^ry-y•^2^r2-^-^^ = o.
Also, the potentials of sources and counter-sources tend to 0 of order •§• as the affix tends to infinity in any direction not asymptotic to ^ , and similarly doublets and counter-doublets tend to 0 of order 3/2 imder the same conditions.
The velocity potentials due to line, surface, and volume distributions in points outside the distributions are obtained by evaluating the integrals fcTiJ * dA,f'3"^ * dS, f ^ x p ^V, where tT denotes the (variable) line surface or volume density, and a> p denotes the velocity potential of a source of unit strength the coordinates of whose origin coincid'e with the variable (s) of integration. For sufficiently regular dist rilrut ion .functions (S" , (e.g. , if Cf has continuous b-,ouadald first derivatives) these integrals exist as ordinary .improper integrals. For instance, for a surface distribution
we obtain
cTdS
^(x,y,z) = r p =
(36)
(x-x;)Hri^[(y-yxY-^ey_
where O' is defined as a function of the parameters u.
the surface S, given by x = y (u,v), z = z (u,vj V , and
of
0 " o • o othe integral xs- taken^<yver those parts of the surface for which
and x\ X .
the i n t e g r a l i^ taken^<yver thos
.-15-The position i s different in the case of
l i n e , surface, or volume d i s t r i b u t i o n s of doublets, since
the i n t e g r a l s corresponding to such d i s t r i b u t i o n s , v i z . ,
T ö i ^ p ^-^J \^ f % ^S, a n d f c ^^'^ dy are in general
i n f i n i t e . Thus? for a surface d i s t r i b u t i o n of doublets
whose axes are a l l p a r a l l e l to the z-axis we obtain
r -tr/"z-z^A^ dS
A 2H - „ (37)
3/2
^ ^
J[(-K)'-^^[(^-^oy<-^Vl]
which is, in general, infinite. However, _pro-videdcy is sufficiently regular', the finite part of the integral still exist.s, and we may say that
^-''jcr $ ^Y^S(curU)
is the potential due to a doublet distribution over S, (with similar definitions for potentials due to line or volume distribution. . An alternative method which avoids the use of the finite part and which has been used by
Schlichting (ref,2) and others, is to consider first the corresponding integral for sources (eqn.(36)) and then to differentiate \rLth respect to (-.z). From a physical point of view this means that we calculate the potential due to
two infinitely near source distributions of opposite strength. The final resuD.t is the same since, according to the rules given in the preceding paragraph, finite parts can always be differentiated under the sign of the integral. It is precisely the possibility of carrying out all the necessary operations directly, without fear (or certainty) of obtaining meaningless symbols^which makes the finite part such a
convenient concept. It will be seen that the alternative method is applicable only when all the doublets have
parallel axeso
In order to define the potential due to a volume distribution of sources (or of counter-sources) we have, as in classical theory, to take recourse to a limit process, as the intsgpandi?'^ p tends to ao of the order I on approaching the point for which the potential is calculated. We therefore surroiond the point by a small sphere of radius €,; evaliiate the integral excluding the interior of the sphere, and then let C tend to 0. For finite 6 , the integrals in question exist as ordinary improper integrals, and the
limit exists, as 6, tends to 0, since the volume of the sphere tends to 0 as € 3,
V/e are now goiag.to show that the
c-flow-defined as a finite part - across a closed surface surrounding a soi;irce of strength <f is equal to - 2 TTcc,
We have to prove
J = - r gradh $ p dS = 2Tr«T'
(38) where $' is defined by (33).We may simplify the problem without loss of generality by assuming x = y = z = 0 . Now let S be a small cylindrical surface bounded*^by two planes x = ±ck
and by the cylinder.,,ipi y2,^z2= ^2 where ^ > ^ • Then the integral of (35) vanishes everywhere on S except in the circular area belonging to the plane x =o(. • Hence •*J reduces to
^% ^V CfoCfS^dydz
and therefore- ^J = 2TTcr, by (8). This confirms the theorem for the particular case of a circular cylinder,
Next, let S be an arbitrary surface
surrounding the source, then we may find a small cylindrical surface S" of the above description inside S ^ d we only have to show that ^V gradh5p dS = ^f gradh$T>^S.
Js "^ - Js',
^^ '
Let y be the volume bounded by S and S; Then by the divergence theorem for finite parts, (9)»
St/- X dy
f gradh ^ p ^S - T gradh^dS » f div gradh^
J S ^ 7S' ^ JY
P'
But
[
r div gradh$p dy = f^-h^p dy = 0, since $ satisfies
Jv ^ Jv
25) and V does not include the origin. This proves that 38) is-true generally,
Similarly, we obtain for counter sources whose potentiall|''p is given by (34).
gradh Y P dS = 2 T T ^ (39)
I
-17-More generally, if a surface S surrounds a finite number of sources of strengths O'^ superimposed
on an arbitrary field of flow which is regxilar inside S, then
*r gradh f dS = - 2 TT H o ^
•Jo.-(40) There is a similar theorem for counter sources.
In fact, (40) follows immediately from (29) and (38). We may also deduce, taking into account (i), at-^he-. end -oi para. 2, that the c-flux across a surface surroi^nding a doublet vanishes, and more generally that the flux across a closed surface is not affected by the superposition
of doublets either inside it or outside.
Finally, it follows from (ii) at tho ond oê para.2j that (38) can also be applied to a continuous distrioution of soiirces inside a surface S, so
- f gradh § dS = 2*n'jo-J c;
(41)
where J'ö'is the total strength of the sources enclosed by S (and given as a line, surface, or volume distribution). The same applies in the limit when the distribution is actually bounded in parts by S,
Equation (41) shaws that the finite part of an infinite' integral i^ more than an artificial analytical 43onceptj and that in certain cases it may have a definite physical meaning, _In fact, the product of density and
c-flux, -pj'gradh^dS^ is the linearised expression for the total flux of matter whenever that expression exists, for instance if $ is given by a homogeneous voliome distribution of soiirces over the interior of a small sphere of radius g and centre P inside S, together with -Ux correspo ndiqg to the free stream velocity. If or is the total strength of the distribution, then according to (41), -/^j^gradh^dS = 21Tcr/^ . Thus, 2rrfrp is the rate at •
which matter is produced inside S. Now let € tend to 0, while cf is kept constant. Then ^ tends to -IJx •+ ^
where ^ p is the potential of a source of strength <j^ •^ locateci at P. But (f having been kept constant, it
follows that the rate at whioh matter is produced inside S, and hence, the rate ait'which matter crosses S is still 2fY<s'p» And this, by (38), can be expressed by
-p f gradh § p dS = - p C gradh (-Ux + $ ^ dS,
so that the finite part is the natural generalisation of an ordinary integral when the latter diverges,
Applying (41) to a small surface .surrounding a point P inside a volume distribution of sources, we obtain
_
r
gradh$dS = 2Ttro^dy
and transforming the left hand side by means of the divergence theorem this becomes
div gradh$dV = 2Tr/ö'dV - j div gradh^dV = 2 ^ ( 0
J y Jy Since this is true for an arbitrary small volume containing P, we must have
A h $ . = div gradh'$ = * 2Tr?r (42) which is the counterpart of Polsson's. theorem in subsonic
theory.
Conversely, given the differential equation (42) over a certain region R, a particular solution of it is
# - f o ' f * P ^V (43) R - ^
The general solution, as is^asily s e ^ by subtraction, then is $ = J of 6*p cl'V-+$ where ^ is an arbitrary solution of (25),
G-iven a surface distribution of sources, it can be shown that the com^jonents of the gradient and therefore of the hyperbolic gradient of the potential remain finite and-,
continuous on either side of the surface S. Also, ^ , and therefore its tangential derivatives, are continuous across the surface.
In order to find the discontinuity of the normal derivative across S,' we apply (41) to a small cylinder whose bases are parallel to the surface on either side of it, and whose height is again small compared with its lateral
dimensions (Fig.2), Letting first the height of the cylinder tends to 0 we find that f ^gradh $ <iS - f ^radh £ dS = 2 II { cT dS where S' denotes the portion of S inside the
J S' r-cylinder, and S+ and ^denote the two bases of the cylinder.^ respectively, 3' 'being the base whose outs.ide normal coincides with the outside normal of S. L etting S tend to 0 round any given point on s, we obtain, denoting by A , z*^» V the direction oasine-s of S in the point in question, and indicating by "i the derivatives of $ on the two sides respectively,
-19-M ^ ! ^ )4-v/iiiL- ^^\= 2vr<r
(44) . In particular, if the distribution is in the
X - y plane, we have A=/*- « 0,V = 1 , so that
2 J - I ^ = 2'Tr6- (45)
"^ z , "^
z-t
a result which is of considerable importance for the calculation of t'lo wave dra^, of an aerofoil moving at
supersonic speed at zero incidence i^refs. 4, 5, 6 ) , A similar relation for the discontinuity of the potential across a doublet distribution in the x y plane, and which
can b-e derived from (42), is fundajnental in the supersonic theory^of flat aerofoils at incidence.
These relations have hitherto been inferred by analogy with incompressible theory and then pi^oved ad hoc in the particul.ar cases required. The need for a more systematic development was pointed out in the introduction to ref. 5.
Dividing (44) by A ^ A / ^ • ) " / * ^-*-^^ we obtain the result that the discontinuity of - 2 ^ in a direction n/ whose components (direction cosines) are — A i i , ^ ,J£. is
217"q-". And since the tangential derivatives of $ are continuous across the surface, it follows that the
discontinuity of q;— must >e the discontinuity of ^-^ ,
a n ' ^ -n where n =fAj /^jV) i s normal to S, m u l t i p l i e d liy t h e cosine between n_ and n ' . Hence
11-21 = 2Tr£^(->2/32+/x2+y2)|A or
Equation (46) amends t h e statement i n r e f . 5 t h a t t h e d i s c o n t i n u i t y of the normal d e r i v a t i v e s i s always 2 fr TT The p a r t i c u l a r case i n which t h i s statement was a p p l i e d ,
however, v i z . ( 4 5 ) , remains c o r r e c t .
The chief use to which Hadamard p u t s h i s concept of t h e f i n i t e p a r t i s r e l a t e d to the above a p p l i c a t i o n s but i s the outcome of a r a t h e r d i f f e r e n t approach.
Hadamard's purpose i s the s o l u t i o n of Gauchy's i n i t i a l
value problem f o r a very g e n e r a l c l a s s of hyperliolic p a r t i a l d i f f e r e n t i a l eq.uations i n c l u d i n g (20) as a s p e c i a l c a s e ,
We are going to develop Hadamard's result in respect of equation (20), i.e. we are going to find an expression for the value of a solution S of (20) in a poin+ P = (XQ, JO> Z Q ) inside a closed surface S, when the values of 3^ -d-nd of iS>Y_ are known on S, Y/here for every point of S, the di^'cction n' is defined as above. For this particular case, an equivalent formula had been derived
previously by Yolterra.
Letlp**P be the velocity potential of a counter source of unit strength located at P; then the function-^ U-^''"p ia admissible
inside S, excluding only P, provided § is regular inside S - and this can in fact be verified a posteriori.
Let us surround P by a small surface S' an^, apply equation (32) to the volume V bounded by S + S'. Since ^hf - Ahf-'y = 0, we obtain
'J
(^gradh'l^^p - ^ * p gradh ^)dS = 0 SfS'or, taking the inward normal as the direction of a surface element of S, and the outward normal as the direction of a surface element of S'
"*'r ^ (-J gradh |,r'p -.^-'p gradh ^)dS = *r (^gradh ^^-^p - "Jr^p
gradh ^)dS (47) It can be shown that as S' contracts to the point P, the
left hand side of (47) tends to ZTlCtixQ, JQ, Z Q ) . Thus
2ÏÏjt(xo, Jo, ZQ) = Y s ^ ^ gradh\/r-^ ..,p^ gradh ^)dS (48) Denoting 'bj A ,j^ , V the direction cosines of the normal to
dS, we have, for any scalar function 3 ,
^radh
f
dS e .• (-M-2 èiL- + ^ d ^ + -j^ _aj )dS
' & x: Ö y ^ z'WOT (S = 1 , t h i s becomes g r a d h 9 dS = - -èJa,,, d S , where n* = - "" d n'"
( A , -/<- , - V ) • Hence, f o r / ? = 1
zTTfixo, Jo, zo) = Ys ^f^ | 4 * ' i ^ - ^
" ) as (49)T h i s i s Hadamard's f o r m u l a (58] ( r e f . 1 , p . 2 0 7 ) f o r t h e s p e c i a l case f = 0 . The d i r e c t i o n n i s c a l l e d by Hadamard t h e t r a n s v e r s a l d i r e c t i o n t o d S . I t s
-21-geometrical interpretation, due to Coulon is that it is conjugate to the tangent plane to dS with respect to the cone /x-Xn"^ 2 _. pry-yX 2 ./g-zrN^l = 0, whose vortex
V ^^ IS ^ ^ ^ ^ J is located at dS.
'^y^* Vorticity distributions in steady supersonic flow. We now direct our attention to the study of rotational motion.
Given a field vector g_ = (u •"•; •'•') ,,ve denpte by ?, 11 , t "tiie components of curl q', \ = .Ü_Ü _ A l , 7 = - 2 - l i - ^ ^ , 5 = — ^ - - 2 J i . The differential
ll z "5 X ^ x B y
equation of the system of vortex lines is ~F ~ « ~ "^ ^^ usual, and the strength of a vortex tube is defined as the
product of the cross section cT into the resultant vorticity
^ ~ l^S "^ / J ^ and is the same at all points of a vortex. All thüse results and definitions are in fact quite independent of whether the fluid is compressible or incompres-sible, except that in the case of supersonic flow, it may be necessary to consider the finite parts of integrals of the "type j q d< and [ curl q dS in cases where the ordinary
JC Js "" " integrals do not exist.
Applying the vector operator 't^'h in cross multiplication to a vector q = (u, v, w ) , we obtain a vector which will be called curlh ^ (hyperbolic curl of q ) .
,;.;cpL...citly
-eurlh 2. =(^ -- — i '2ji ^-A^ ^ ,.n.a^^Q}^\ (50)
V B y ^ z 3 z ^ ^ x '^ 'Bx -c^yj Direct calculation shows that
divh curlh g_ = 0 .,,...., (51)
and
curl curlh q. = gradh div 2^ - div gradh ^ .,.,,,... (5?,) 'A field vector q will be called irroiiational or
lamellar, as usioal, if curl £ = 0 , and it will be called hyperbolic-solenoidal if divh g_ ^ 0»
We are going to show that a vector ^ defined in a region R and admissible in it can be represented as the sum of three vectors, one irrotational, one hyperbolic
solenoidal, and one both irrotational and hyperbolic solenoidal, More precisely, we are going to represent £ as
divh q = divh £ , curl q = 0 in R .„o...(53) /(54).,
curl q = curl q , divh q . = O in R ..,.(54)
"2
""2
and
divh q = 0 , curl 3. = O in R ,...(55)
3 3
Assuming that vectors as described in (53) arid
(54) have been foimd, we put ^ = £, - q - Ü • Then divh a =
3 "T- 2 3
divh q divh q divh q = 0 , and curl q = curl q
-~ n. 2 3
curl 2^ - curl £ = 0, so that q defined in that way
satisfies (55).
Putting 6^ = —-== divh q in R, we determine a scalar
function f by ^ « ^ <r $ *p dV , so that £i^ $ = - 2 tr =r
"" JR r- r
according to (42), i . e . , A h ^ = - divh .q, and so a = - grad$
satisfies (53). Thus ' . "^
-1 ^ " 2 T r ^""^^ { ^^'^ 3. • i * P -av ..(56)
To find
2r>
J we shall assume that a is given
as the hyperbolic curl of a vector j£ =ni'-^ ,Y^ > T^) , q«
= curlh V , and so (57) curl q„ =-trurl'curlh T" = gradh *
div;^ -'aiv gradh ^ , by (52)."^ We now restriotjIC
by ahe condition thai*' div Y = 0. Then we must hSve
div gradh"^/= - curl £ = -*curl q , by (54) and (57).,
i.e., S2- 2/Ih
Y = - S , A h Y « - 1 J , A h ^ = - S i r i R ..(58)
and t h i s according to (42) i s solved by
JR - * dy P R o r 35v
Y = - ^ I curl q è * p dy
(59) /Ehen
-23-Then q^ = curlh,]H^ satisfies (54), provided v/e can show that in "^^fact div'V = 0 > as assumed. And this can be shown exactly as in The classical counterpart
(ref.- 7, para.148), provided the integrand vanishes at
infinity or, in particiiLar, if it vanishes outside a finite region. This again will certainly be the case if curl g
vanishes for sufficiently small x, since Ö vanishes for
sufficiently large x, P It is clear from the above construction that q
determines the flow due to the source distribution "1 in R, while q represents the flow due to the vorticity
distribution. "^ Given a three dimensional vorticity distribution we then have in detail
(60) (The expression on the right hand side is actually an ordinary
improper integral), wherQ R' is the sub-domain of R which satisfies ( J - x ^ > ^ 2 r>'y.y^2^ /2.2^J and x^<x. We then obtain for the components u, v, w, of q = curlh
1 " !
"U =jy
air
fe-o-)^-C-yo)^]
^^-^2^^^^-V=-2ir
dx dy dz 0 0 0 - / 3 ^ /• 17 -vb n ^^o ^yp ^=^0 3/2 ' h i s may be w r i t t e n wherea ^ ^ ^ ^ r ^(r X ourl a) ^
r = f x - x , y - y , z - z \ , and \ o Q o) 3/2 (61) (62) s = 2.zj . , .
- ^Tllie corresponding^ £oojmula---jEor--iruK>mpressible flow i sa_' = r (r X curl q') ÉI
r5
R' ^ ^. (63)
The discrepancy in sign is only apparent, since as, for instance, formula (8) shows, the sign of a finite part does not follow the sign of the integrand, as for ordinary integrals.
We may now calculate the field of flow due to an isolated re-entrant line vortex C. Replacing the volume element dx dy dz in (60) hyrS'^djlQ where d%Q is the
0 o Q
element of length of C, and CT* its infinitesimal cross section, and writing <^ =V^S "+" "^ "^ 5 / "*, we have
dx cly dz^
| = ^ " j f - ' ^ = ^ 1 ^ '^ = *^"ir"
' ^^ ^°'
since CO(r^
is a constant K, and since d^ = f dx , dy , dz ) , (60)0 V o o 0/ beoome-s
Y{x,y,z)=j|p
f tL
-(64)
where C' consists of the segments of 0 which satisfy
If in particular C consists of straight segments which are parallel either to the x-axis or to the y-axis, then
(64) can be integrate'd, since
dx^ -, X - X , --L o J. c o n s t , = - cosh T and
^t
dy^ • 1 - 1 / ^ ( y - y p " ) 4. . ^ o = « — s i n ^ '— T* c o u s v . / 3 >•--;)' -fi'p'^^-14] X^"^' '^ c^-^^'
(65) Now l e t , 0 be a ' h o r s e s h o e v o r t e x ' of s t r e n g t h K, oonèis-tiog of t h e s . t r a i g h t segments C x , < x < ' o o > y =-!•.* >'.-'• ^ \0 o 1 z- = 0> , fx = X-, - ' y < y ^ y- , z • = o"^ and o ^ V o l l o l o ^ / ' x < x < ' c x ) , y = y > 2 = O) ^tihevQ x and y a r e " v i ^ o o l o / 1 1 g i v e n c o n s t a n t s ( F i g . 2 ) . U s i n g ( 6 5 ) , we f i n d t h a t 3 1 ^ 'T]/ = 0 a l w a y s , and llr ='Y = 0 f o r x < x ^ , w h i l e f o r x > x ^ , / ( 6 6 )-25-9 TT L
-1
X-cosh
^1
- oosh
-1
X - X1
/w^
2 ^
2 ^ ^ /^"^^O^TT
-1 ^ ' ^ (66)
where the cosh are to be replaced by O when their
arguments are smaller than 1, respectively, and
•Y' = --^
-=— I s i n -v * >=-^'- —n
. -1 /3 C y-yvi
. - i / 3 ( ^ - y { )
x2 />2 2 ,
(67)
where the sin are replaced^_iL or -JJ- when their
r 2 2
arguments are greater than 1, or smaller than - 1 ,
respectively.
We now obtain q_ = (u,v,w) by taking the
2
hyperbolic curl of^f =rY-'-
ij^
^ O) , so that
u = V = w = 0 for x ^ X y while for x>x-,,
u =
K/T
2'iT
«^ ( ^ ' ^ ^ • ViL_[ - ^
2-tr ^-7-;^^^^^^^^^'^f'i^^yfTT^
W =K
2-^
[.
(^x-xj.-) (y-y;;) £ x - x ^ -/j'[(y-y^% ^ ^ J
(x-x,-^(y^y^[(x-x^ - ^ ^ f ^ y ^ ^ ^ 2.^]]
(68)
/where ,,..
where, for given x, y, z, the imaginary terms are omitted, Except for the notation, equations (68) agree with the field of flow round a horse who e vortex calculated by 'Schlichting l^y an entirelydifferent method (ref,2)
Some care is required when attempting to represent a volime or surface distribution of vortices as a combination of line vortices. Thus, according to (68), the components u , V, w, all vanish when (x. y,z) is outside both the
cones (^-^J "/3^R^y+ yïf-+ ^ J emanating from the tips But it can >e shown that this is no longer the case if the vorticity in the spanwise segment is distributed over a finite
width A •^ However, even then the failure (which is due to the discontinuity of 'yjT for line vortices) can occur only at points belonging to tffe envelope of the cones of type
2rr „-s? . /„ . A 2 )
(^ - ^f-ft%y-^^^ (-- ^^
= 0 emanating from thevortex lines which are supposed to generate the surface or volume..
Applications to aerofoil theory are g,iven in a separate report (ref.8).
2 7
REFERENCES
-Author J , Hadaxiard
Title, e-Qc.
Lectures in Cauchy's problem in linear partial differential equations.
New Haven - London, 1923. H. Schlicliting Tragfiugeixheorie bei ^
Uberschallge schwindig^keit LUi. tfahrtforschung, vol.13, 1936.
R, Courant and D. Mlbert.
Methoden der Mafhematischen Physik, vol.2, Berlin, 1937. A. Robinson The wave drag of diamond shaped
aerofoils at pero incidence. R.A.E. Tech. Note No.Aero
1784, 1946.
A. .Robinson
The wave drag of an a e r o f o i l
a t ' z e r o i n c i d e n c e .
1946.
A,E. Puckett, Supersonic wave drag of thin airfoils.
Journal of Aero. Science, vol.13, 1946.
H. Lamb Hydrodynamics, 6th vol, Cambridge, 1932.
A. Robinson and J.H. Hun-feer-To-d
Bound and trailing vortices in the linearised theory of supersonic flow, and the downwash in the wake of a Delta wing.
College ofj Aeronautics Report No.10, 1947.
CALCULATION O F A FINITE PART
H
f
F I G . 2
CALCULATION OF THE F I E L D OF FLOW O F A S U P E R S O N I C H O R S E S H O E VORTEX.
