OF SHOCK WAVES*
4. The properties of the “limiting lin e.” Being the envelope of one fam ily of real
354 YUNG-HUAI KUO [Vol. IV, No. 4
B y continuity, the relative signs of the differential quotients hold in the neighborhood of the “lim iting line.” Thus, we conclude that either v t > 0 , yx> 0 and « , < 0 , uz < 0 or V i < 0 , vx < 0 and « ¡ > 0 , u x> 0 . The first case is exactly the condition for a compres
sive motion. Whereas the second case may either correspond to a rarefaction or to a change of sign of the Jacobian J (u , v). As the rarefaction does not conform to the geometric properties of 7 = 0, the second case corresponds to the second branch of the solution and hence can be disregarded.
5. Lost solution. In the previous sections, we assume that the Jacobian J(n, v) does not vanish. Thus the one-to-one correspondence between the t, x- and u, ti-planes is assured and the condition 7 = 0 is restricted to the singular line /. In a special case the Jacobian may vanish identically, however. T his vanishing of the Jacobian estab lishes a relation between v and u in the u, u-plane and, as a result, yields a class of solution not contained in the transformation (2.7). To study this form of solution, let us first set
T his type of solution has been discussed by K. Bechert7 whose main result was as follows. By elim inating x and t the system of Eqs. (5.2) and (5.3) can be reduced to a second order non-linear total differential equation, based on the existence of a linear
type being amenable to numerical integration. The main feature of the solution, how
ever, can be discussed in the following manner.
Along ii = const., i.e., along
on account of Eq. (5.2). Since d v /d it is a function of u alone, on u = const. (d v / d u ) u is
V = v ( u ) . (5.1)
T he differential equations (2.4) and (2.5) can then be rewritten as / d v\
u t - f I ■u + — j u z = 0, (5.2)
(5.3)
relation between t and x. By a slightly different procedure it can be shown that in
stead of a second order differential equation one can obtain a first order one of Abel’s
du = iixdx + Htdt — 0, the slope of the curve u = const, equals
(5.4)
7 Bechert, K., Über die A u sb reitu n g von Z ylin d er- u n d Kugelw ellen in reibungsfreien Gasen u n d F lüssigkeiten, Ann. Phys. (5) 39, 169 (1941).
356 YUNG-HUAI KUO [Vol. IV, No. 4
constant. Therefore, the curve « = c o n s t. is a straight line in the t, x-plane. In con
form ity to the assumption (5.1), there exists a parameter £ defined by
£ = - . (5.5)
Co(t -f- to)
where c0 is the speed of sound at u = 0, and to a suitable constant. It is clear that
£ = const. corresponds to « = const. In other words, both v and u may be regarded as functions of £.
If the determ inant v'2—pv?£0, it, and u t can be expressed in terms of u. We have
a3uv 1
« > = -71— 7F ’ (5 -6)
X v 1 — pv aBuv u + v'
M‘ = X ~n v — pvI T ’ (5 ' ?)
where the prime denotes the total differentiation with respect to u. Like in the gen
eral case, here again the solution possesses a singular line on which the partial deriva
tives generally become infinite. Its other properties will be studied presently. From Eq. (5.4) it is found that
Q . - + *
while the characteristics are
( ~ ) - «
\ d t J c +
On the other hand, where the singular line X, i.e. the line
v'2 — ¡3v = 0, (5 .8 )
intersects the integral-curve v(u), we have
Q - ”±v®-(IX- (5-9)
T his shows that at the singular point of the solution v(u), the « = c o n s t. line be
com es the envelope of one fam ily of characteristics C. H ence the envelope is a straight line. Furthermore, according to Eqs. (4.1) and (4.2) the parametric equations of the path 5 are
dx = —— (v' + \/ f} v )(v ' — \Z fiv )d u , (5.10) ajW
dt = — (n' + V Ñ W ~ y/f}v)du. (5.11) afiuvv'
Since each factor on the right-hand side corresponds to a group of the characteristics C, on crossing the line X, where this factor vanishes, the elem ents dx and dt change
their signs. T his proves th at the line I, the image of X, possesses all the characteristics of a “lim iting line.”
It is interesting to note the difference between plane and spherical waves. In the former case, Eq. (5.8) would be satisfied identically. This lets the lines u = const, de
generate into the characteristics. Indeed, it is also possible for one fam ily of the char
acteristics which are straight lines to have an envelope; the differential quotients u x, u t are finite, however. Consequently, we have no “limiting line,” in the strict sense.
This docs not mean, of course, that the solution is regular. As a matter of fact, the solution already becomes many-valued before this line is reached.
6. Lost solution: a special problem. From the foregoing conclusions, a com pres
sive spherical or cylindrical wave always becomes indeterm inate when a singular line is reached. As an illustration the following special problem is considered.
Suppose there is a divergent spherical or cylindrical w ave propagating w ith veloc
ity c0 into still air. On the wave-front, where the motion agrees with the outside con
ditions, the state-variables p, p become equal to those of the still air and the velocity is zero. T he path of the wave-front is then described by
are propagated with constant speed. In other words, these quantities depend only on a common parameter.
T o sim plify the am ount of mathem atical work involved, the differential equations (2.4) and (2.5) will be put into the following equivalent form:
In the case of a lost solution, there exists a parameter £ defined by (5.5) such that
£ = 1 corresponds to the initial curve (6.1). Then, x — co(t T to).
T h e mathem atical problem can thus be formulated in the following way:
u = 0, when x 2: c0(t + to), | u 0, when x < c0(t + to) - )
( 6 . 1)
(6.2) A particularly simple case will be the one where both the pressure and the velocity
2 cV»
(c2 — <t>x)<t>xi — 2<t>x<j>zt — 4>ti 4— - — = 0 (6 .3 ) x
by introducing a potential-function </>(/, x):
u = 4>x, (6.4)
<t>(t, *) = cl{t + /o)/(£) (6 .5) and hence
u(t, x) = c0/'(£),
c2 = d l i - j r - K f - £ / ' ) ] ,
(6.6)
(6.7)
where the prime indicates the total differentiation with respect to £, and the function /(£ ) satisfies
358 YUNG-HUAI KUO [Vol. IV, No. 4
[c2 - d ( f - ay]af" + 2e l f = 0 (6 . 8) subject to the initial conditions
/ ( I ) = 0, / ' ( l ) = 0. (6.9) T he first condition, n a m e ly /(l) = 0 , is necessary to make c = Co on £ = 1. When the conditions (6.9) are substituted in Eq. (6.8), it appears t h a t / " ( l ) is arbitrary. We need not be alarmed by this situation, but recall that in this particular type of initial value problem, the “support” is a characteristic. Physically, this means th at the ini
tial conditions prescribed in this manner do not “know” the internal structure of the m otion, because they propagate ahead w ith larger speed. It is only natural, then, that such an arbitrariness should arise which enables us to fit properly the physical condi
tions specified. T his arbitrariness is only a partial one, however, since for a compres
sive motion the sign o f / " ( l ) is necessarily negative; for on £ = 1
Thus, for any com pressive motion the absolute value of / " ( l ) is determined in con
sistence with the physical process.
T he differential equation (6.8) which determines the interior motion of a mass of air, has two singular points in the £, /-p lane given by the vanishing of the coefficient o f /" (£ ). T he geometrical interpretation is evident, when (6.8) is written as
( c + « — c o £ ) ( c — m - f- c 0£ )
that is, when one fam ily of characteristics become tangent to a line £ = const., an infinite curvature would occur if u is finite there. According to w hat has been said in the last section, this characterizes the “limiting line” of the solution.
Let us push the discussion a step further. For this purpose only the first order terms need be retained. Taking j3 as a small parameter, one has accordingly
(pi) i + P o (w i)i = 0,
according to Eq. (2.2). In a compressive motion (p ,) i> 0 , it follows that
(« * )i = - A D < o. (6 . 10)
X
(6.11)
(6.13) (6.12)
— = --- , 0 g £ g 1.
d£ £ («, — £) =— ! (6.14)
Aside from the two singular lines
w = £ + 1, (6.15) w = £ - 1, (6.16)
w here th e slope of w is infinite, there are tw o ad d itio n al singularities (1, 0) an d (0, 0)
360 Y U N G -H U A I K U O
back to th e t, x-plane, will correspond to th e very one t h a t doubles back a t th e “lim it
ing lin e.” A co ntin u o u s solution is th u s o u t of th e question.
T h e a lte rn a tiv e p rocedure would be to co n tin u e it b y joining it sm o o th ly a t the line X to th e lost solution. T h is is also im possible. Ind eed, if th is were possible, th e line X w ould hav e to coincide w ith th e integral cu rve v(u) in o rd e r to prov id e a co n
tin u o u s solution. T h is is co n tra d icto ry , because it is easy to show t h a t th e line X does n o t satisfy th e differential eq u a tio n for v ( u ) .
T h e o th e r possibility which rem ains to be in v estig ated is to identify'- th e “lim iting line” as a shock w ave so as to c o n s tru c t a disco n tin u o u s solution. T h is w ould require th e contin u ed solution to satisfy th e shock conditions. Since, in general, th e “lim iting line” I is curved, as a re su lt th e re would be a non-uniform shock w ave in th e m otion, for w hich b o th th e speed and th e s tre n g th are no longer c o n sta n t and therefo re th e e n tro p y w ould be c o n sta n tly ch anging across th e shock. T h is very fa c t m akes th e original assu m p tio n u n ten ab le . H ence to co n tin u e d isco n tin u o u sly a solution w ith en tro p y c o n s ta n t everyw here is also im possible.
T h e problem m ig h t be solved, how ever, if th e original hy p o th esis of isentropic m otion is ab an d o n ed . T o include th e possibility t h a t a shock w ave m ay ex ist w ith in th e m otion, th e co n tin u ed solution m u st satisfy th e following m ore general set of e q u a tio n s:
u, + u u x + — = 0, (7.4)
P
( a u \
P i+ « P i + p ( « x H— -— ) = 0, (7.5)
(pP~y)t + u(p p -y)x = 0. (7.6)
T h e ta s k th e n is to c o n stru c t a solution w hich should satisfy b o th th e in itial an d th e shock co nd itions in a region bounded b y th e initial curve, th e shock line an d a ch ar-, acte ristic d raw n to th e initial curve th ro u g h th e p o in t w here th e envelope first a p pears. T h e shock line, how ever, is n o t given, it should be chosen in such a w ay th a t it yields a solution fulfilling all th e prescribed conditions. T h e m a th e m a tic a l problem th u s tu rn s o u t to be extrem ely difficult.
T h e a u th o r w ishes to th a n k D r. H . S. T sien for his in valu able discussions and criticism .
O N P R O J E C T IL E S O F M IN IM U M W A V E D R A G *
BY
W IL L IA M R. SE A R S Cornell University**
1. Intro d u ctio n . T h e w ave resistance of slender bodies of revolution in sy m m e tri
cal supersonic flow w as calculated ap p ro x im ately b y von K â rm â n ,1 b y m eans of a d istrib u tio n of singularities along th e axis of the projectile. T h e in divid ual sin g u larity is ch aracterized by a p o ten tial of th e form <£,(x, r) = j (x — £ .)2 — a 2r 2}-!/2, w here a;, r are cylindrical coordinates, x being m easured d o w n stream from th e nose of th e p ro jec
tile and r radially from th e axis, is th e value of x corresponding to th e sing ularity , a is the co tan g e n t of the M ach angle of th e u n d istu rb ed flow, so th a t
a = V ( u / a y - 1,
U and a being th e stream velbcity and th e velocity of sound in th e u n d istu rb ed flow.
I t will read ily be verified th a t <£j(x, r) is a solution of th e linearized p o ten tial eq u atio n for supersonic flow w ith axial sy m m etry
/ U2 \ d2<f> d24> 1 d<}>
( 1) — - = — + — — • (1)
\ a2 ) dx2 dr2 r dr
Von K arm an calculated the w ave resistance by in teg ratin g the tra n s p o rt of m o
m entum across a cylindrical surface enclosing th e body. In his app rox im atio n, the integral is in d ep en d e n t of r and can be ev alu ated in the lim it r —>0. T h e re su lt isf
R = - irp f f /'(* ) /'(£ ) log | x - £ | dxdÇ,
J 0 J o (2)
w here R is th e w ave resistance and /( x ) is the fun ctio n specifying th e d istrib u tio n of singularities along th e x axis. F o r bodies of finite length / ,/ ( x ) is found to be in den ti- cally zero for x > l ; hence b o th integ rals in (2) can be replaced b y in teg rals from 0 to I.
F o r slender bodies, von K arm an showed th a t app ro x im ately U dS
M l = 7 7 ! ( 3 )
Z7T ( I X
where 5 is th e cross-sectional area of th e body.
In th e p resen t p ap e r we shall am plify the analogy, alread y m entioned b y von K arm dn, betw een th e w ave resistance of a slender projectile and th e induced d rag of a wing. I t will be shown th a t th is analogy suggests a useful form for the calculation of
* R eceived June 11, 1946.
** T h is w ork w as undertaken w hile the author w as em ployed b y Northrop Aircraft, Inc.
1 T h . de KArmdn, The problem o f resistance in compressible flu id s, A tti del V C onvegno della “Fonda- zione Alessandro V o lta ,” R om e, 1935, pp. 2 2 2 -2 7 6 .
t Von K d rm in 1, Eq. (9.12). I t m ight be m entioned th a t th is formula is m ost easily obtained from E q. (9.11) of th e sam e reference b y first integrating b y parts w ith respect to x in order to obtain a form sym m etrical in x and £; it w ill then be found th at a double integral carried over half the first quadrant of an x, $ plane can be identified w ith half th e sam e integral carried over th e entire quadrant.
362 W IL L IA M R. S E A R S [Vol. IV , N o. 4
T h e induced-drag analogy pointed o u t above suggests t h a t / ( £ ) be expanded in a
364 W IL L IA M R. SE A R S [Vol. IV, N o. 4
should be m entioned th a t this com parison m ay be m isleading in view of th e fa ct th a t von K d rm a n ’s ogive has a b lu n t stern , so t h a t its w ave d rag certain ly does n o t re p re
sen t its e n tire resistance, even in th e absence of skin friction. N evertheless, th e w av e resistance of th a t ogive m ay be tak en as a conven ien t reference.)
T h e shape of the forw ard half of th e sym m etrical projectile represented in (13) is draw n in Fig. 1, for th e case l = 4 d max. F o r com parison, th ere is also show n th e shape of von K a rm a n ’s ogive having the sam e caliber and one-half the length.
Fi g. 1. Profiles of various projectiles of m inim um w ave drag: (a) volum e and len gth given , (b) caliber and length given , (c) von Kdrmdn’s ogive of equal caliber and one-half th e len gth. (P rojectiles (a) and (b) are sym m etrical fore-and-aft.)
4. M inim um w ave drag for given caliber an d leng th . T o a tta c k th e problem of the body shape for m inim um w ave drag, caliber an d length specified, we re tu rn to th e expression for th e w ave d rag given in (2') an d (4) an d em ploy th e m eth o d s of v a ria tion calculus. By v irtu e of th e sy m m etry w ith respect to x an d £, th e v aria tio n of th e resistance w ith v ary in g body form assum es a sim ple form ; viz.,
oi? = - xP | J ‘ Sf'(x) J / ' « ) log | * - £ | d & x
+ J / '( £ ) J $/'(£) log | x - ? |
= - 2xp f 8/ '( x )F ( x )d x . (15)
j 0
In th is section we shall provide for th e possibility [excluded in o b tain in g ( 2 " ) ] th a t d S /d x , an d th e re fo re /(:*;), is discontinuous a t th e sta tio n w here th e m axim um d ia m e te r occurs, x = m. H ence, in te g ra tin g by p a rts in (15), an d again assu m ing sh arp po in ts a t bow and stern , we w rite
5R = — 2 t P | f ( » ) 5 [ / & ) ] „ -
J
8f ( x ) F '( x ) d x ^= - 2rp | F ( « ) i [/■(*)]„ + L- f 8S ( x ) F " ( x ) d x j , (16)
where [f(x)]m denotes the value of the d isco n tin u ity in f ( x ) a t x = m, and th e area function S(x) has been assum ed to be continuous.
In th e form (16) it is clear th a t th e shape of th e p a r t of th e body forw ard of the m axim um section a t x = m can be held fixed while S(x ) is varied o ver th e rear p a rt to achieve a m inim um of R ; th en th e re ar shape can be fixed in th is m in im um -d rag configuration while S(x ) is varied in fro n t to m inim ize R-, th e re su lt will be th e m ini
m um -drag shape for given m axim um cross section a t x = m. W e shall also assum e th a t th e d isco n tin u ity of slope represented by [/(x)]m is n o t varied in th e process; it will a p p e a r la te r th a t th is is valid. T h e m inim um -drag condition 8R = 0 is th en obtained w hen
F"(x) = 0 )
> 0 ^ x m, F(x) = cix + c2)
(17) F "(x) = 0 )
>m £ x ^ I.
F(x) = c3x + c j
T h e analogy w ith the induced d rag of a wing is again useful. T h e analogous p ro b lem is th e following: to d eterm ine th e spanw ise circulation d istrib u tio n f ( x ) so as to o b tain m inim um induced drag, it being required th a t th e to tal lift be zero, b u t th a t th e lift carried on one side of a sta tio n x — m have a given value, equal and opposite to th a t carried on th e o th e r side of th a t statio n . T h e re su lt o b tain ed in (17) s ta te s th a t th e condition of m inim um d rag results w hen th e dow nw ash F '(x) is c o n s ta n t in each of th e two p a rts of the wing.
F o rtu n a te ly , inv estigations have been m ade4’6 of the b ehavio r of th e circulation d istrib u tio n n ea r a p o in t on a lifting line w here the dow nw ash in discontinuous. I t is found th a t th e circulation function is continuou s b u t has a vertical ta n g e n t an d d is
con tinuous c u rv a tu re a t such a point. A pplying th is re su lt to o u r projectile problem , we can conclude th a t /(x ) will exhibit a sin g u larity of this ty p e a t x = m. M oreover, since F(x) can be in terp re ted as th e dow nw ash corresponding to th e circulation dis
trib u tio n S ( x ), we conclude th a t F(x) c a n n o t be discontinuous a t x = m if we exclude singularities of th is ty p e from th e shape function S(x). A ccordingly, we w rite
(ci — c3)m - Ci — c2. (18)
T h e resistance of the m inim um -w ave-drag bo dy is easily calculated from (2 '); it is
4 A. B etz and E . Petersohn, Z u r Theorie der Querruder, Z. angew . M ath. M ech. 8, 253-257 (1928);
also N a t. A d vis. Com. for Aeron. T ech . M em o. N o . 542 (1929).
5 H . M ulthopp, D ie Berechnung der Auftriebsverteilung von Tragflügeln, Luftfahrtforschung 1 5 ,1 5 3 — 169 (1938).
366 W IL L IA M R. S E A R S
7 W . H aack, Geschossformen kleinslen W ellenwiderstandes, Bericht 139 der L ilienthal-G esellschaft fiir L uftfahrt.
T H E B O U N D A R Y L A Y E R IN A C O R N E R *
BY
G. F. C A R R IE R B row n U niversity
1. In tro d u c tio n . T h e lam in ar flow of a re la tiv e ly non-viscous fluid th ro u g h a c h a n nel is ch a rac te rized by th e presence of a th in b o u n d a ry lay er along th e w alls. In s tra ig h t channels, such b o u n d a ry layers are u su ally assu m ed to h av e th e velocity d istrib u tio n d eterm in e d b y B lasius [l ] for th e flow p a s t a flat p late, a n d th e flow p a tte rn in th e neighborhood of a n y corner is n o t m entio ned . I t seem s of in te re st to develop here th e change in th e B lasius flow im plied b y such a corner.
2. T h e b o u n d ary la y e r problem . W e shall consider th e la m in a r flow of an incom pressible fluid w hich im pinges w ith th e uniform v elocity V on th e edges # = 0 of th e half planes y=*0, z = 0.
T h e N a v ier-S to k es e q u a tio n s a n d th e c o n tin u ity con ditio n w hich govern such flows are
(v-grad) v + p~l grad p = vAv, (1)
div v = 0. (2)
H ere v is th e velo city w ith com p o n en ts u, v, w; p is th e pressure, v th e k in em atic viscosity, a n d p th e d en sity .
As v. K a rm a n h as po in ted o u t [2], th e essence of th e tre a tm e n t of such eq u a tio n s in a b o u n d a ry lay er problem is to elim in a te higher o rd er term s (b y a p e rtu rb a tio n schem e or otherw ise) in such a m a n n e r th a t th e o rd e r of th e e q u a tio n s is n o t decreased.
In th is w ay no b o u n d a ry co n d itio n s need be relaxed. W e m ay accom plish th is b y using w h a t is essen tially P r a n d tl’s co o rd in a te tra n sfo rm a tio n [ l ] , n am ely
t, = y / { v x / V ) 1'-, f = z/ { v x / V y > \ (3) W e also define th e p a ra m e te r £ = (v/ F x)1/2.
Since th e flow b o th w ith in an d ou tside th e b o u n d a ry lay er m ay be exp ected to be essen tially in th e x d irec tio n a n d slow ly v a ry in g in x, we m ay a tte m p t to find a solu
tion in th e form
■11 = V[u0(t], f) + £î<i(t), f) + £!M2 + • • • ] (4)
v = F(£»i + £2fl2 + • • • ) (5)
w = V(£wi + £2i£>2 + • ■ ■ ) (6)
P = p V i (p0 + £/>i + • • • )• ( 7)
W e com m ence th e series for v a n d w w ith a term of o rd e r £, because we w ish a so lu tio n for w hich v / V , w / V , are sm all. F u rth erm o re , if we includ ed term s »o, w 0, th e following s e t of eq u a tio n s w ould co n tain term s of o rd e r £-1 w ith no c o n trib u tio n from th e viscous term s of E qs. (1) a n d (2). T h u s th e so lution s w herein Vq, Wo w ere n o t id en tically zero w ould n o t prov ide re su lts corresponding to th e ph enom enon u n d e r in v e stig a tio n .!
* R eceived Aug. 30, 1946.
t A ctually, th e fact th a t our results c o n stitu te a solution which ob eys the differential equation and boundary con d ition s is sufficient justification for taking r , s œ , 3 0 .
T h e su b stitu tio n of Eqs. (4) to (7) in to E qs. (1) an d (2) leads to th e system
«o , v dpo f dpo
(rid llo /dri + f d « o / d f ) + 'OidUo/dri + W i d U o / d f i
---2 2 dri 2 0 f
/ a 2 a2\
- ( v + i p ) “ , + « " ) + ■ " m
dpo dp\ ( Uo T dvi a»i1 a2vi \
V + i 1 7 - i , ( T L ’ V + f 1 7 J + l ? + ' • ) + • • • - ° w
” + + - ° (10)
d u 0 t du o d v i d w i
1 — + {(■••) + ■•■ = o. (ii)
2 dr) 2 d £ dr] d £
T h e solution of th is system of eq u a tio n s requ ires th a t th e coefficient of each pow er of £ in each e q u a tio n vanish. T h e first ord er ap p ro x im a tio n to th e re su lt is defined by th e v anishing of th e coefficients of £°. T h e re su lt can be ex p ected to be valid o n ly w hen the rem ain in g term s of th e series are negligible, t h a t is w hen £ is sm all. T h u s th e solution, like t h a t for th e flat p late, is valid only a t sufficiently large d istan ce s from
T h e solution of th is system of eq u a tio n s requ ires th a t th e coefficient of each pow er of £ in each e q u a tio n vanish. T h e first ord er ap p ro x im a tio n to th e re su lt is defined by th e v anishing of th e coefficients of £°. T h e re su lt can be ex p ected to be valid o n ly w hen the rem ain in g term s of th e series are negligible, t h a t is w hen £ is sm all. T h u s th e solution, like t h a t for th e flat p late, is valid only a t sufficiently large d istan ce s from