Herman Weyl.

Shock waves in arbitrary fluids; a note submitted to the Applied Mathematics Panel, NDRC online

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M „ c/ AM P Note No. 12

Copy No. 5/

AMG-NYU No. 46



^TITUTEof;

FENCES

25 WaVCHy P ^< New Y orh 3f N> y

SHOCK WAVES IN ARBITRARY FLUIDS

(Co M+s^ li^l)



WITH THE APPROVAL OF THE OFFICE OF THE
CHAIRMAN OF THE NATIONAL C.'i F. •'; F. RESEARCH
COMMITTEE, THIS REPORT HAS bEEN DECLASSIFIED
BY THE OFFICE OF SCIENTIFIC RESEARCH AND
DEVELOPMENT.



A Note Submitted
by the
Applied Mathematics Group, New York University

to the
Applied Mathematics "anel
National Defense Research Committee



•"" _ ., II I ,.uv UAMNEI



«-»»»; „.*u March 1944






HAT10NM ut"



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AM? NOTE NO. 12

This note is of a theoretical character. Its results are not
meant to lead to immediate applications; however, the subject
is closely conne cted w ith topics that have arisen in connec-
tion with AMP Study 38 on shock waves.

DISTRIBUTION LIST '

No. of
copie s



2 Office of Executive Secretary, OSRD

10 Liaison Office, OsRD

1 Att: B. S. Smith, British Admiralty Delegation

3 Chief, Bureau of Ordnance

1 AttrR. J. Seeger

1 " P. Keenan

1 " R. S. Burington

4 Aberdeen Proving Ground, Ordnance Research Center

1 Att: Col. L. E. Simon

1 " Hans Lewy

1 " 0. Veblen

1 " S. Chandrasekhar

1 G. B. FistiaVowsky, Chief, Division 8, NDRC
Att: H. Be the

1 Warren Weaver

1 J. G. Kirkwood

1 A. H. Taub

1 J. von Neumann

1 H. Weyl

1 T. C. Fry

5 R. Courant

1 E. J. Moulton
1 S. S. Wilks
1 M. Rees



TA BLE OF CONTENTS

n age

Summary II

Figs. 0, 1, 2 IV

Figs. 3, 4, 5 V

Figs. 6, 7 VI

Par t J. Thermod ynamics and th e Shock phenomenon 1-19

1. Shocks 1

2. Thermodynami cal assumptions 4

>

3. The fundamental inequality for the direction Z Z, - 9

4. Entropy and parametrization of the Hugoniot Line - 12

5. The limit of pressure along the Hugoniot Line 15

6. The example of the "deal gas 16

Part II. Reflection of a Shock Wave 19-29

7. The algebra of reflection 19

8. Why glancing incidence is impossible 24

P art III. The Problem of the Shock Layer 30-41

9. Formulation of the problem 30

10. Character of the two singular points Z , Z, :

l T ode and saddle 34



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II



SUMMARY



The following investigation trios to clear up the
general hydrodynamical and thermodynamical foundations of the
shook phenomenon. The main results are contained in Part I,
.§§1-4, and Part HI, §§9-10. The first part answers the
question: What are the conditions for the equation of state
of a fluid under which shocks with their distinctive quali-
tative features may be produced. Those conditions, enumerated
in §2, are partly of differential, partly of global nature*
Port III investigates the physical structure of the shock
layer whose "infinitesimal" width is of the order of magnitude
t rovided heat conductivity and viscosity are small of the
same order. Initial state and final state are singular points
for the differential equations of the shock layer, and it is
shown that they are of such a nature as to make one expect j "
problem to have a unique solution.

§§5-8 are only loosely connected with the main issue
The author's previous note, ''A scheme for the computation of
ock waves In gases end fluids", dealt with ideal gases for
which the specific heat is not constant* $6 describes how
the two basic theorems ( H, ) and ( H- ) of that paper follow from
the present more direct and more general argument. The prob-
lem of reflection treated in Part II, §§7 end 8, is not of
primary importance for a theoretical understanding of t]
mechanism of shocks, but is of t practial importance.

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It is shown hero that the algebra of that problem is entirely
independent of the underlying thermodynamics (as is also
abundantly clear from von Neumann's studies), and this alge-
bra is brought into what I believe to be its most satisfactory
mathematical form (by using 6" = tan which are
divided by the shock front x = 0. Because of conservatioa

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f

. J . . .



-2-



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of mass, P u = fi u i » an< ^ denoting by M this common flow
of mass, we may write

o o' 1 1
Then the laws cf conservation of momentum and energy give

the further relations






Mu 1 + p lf


Mv = Mv, ,
o 1'


Mw - Mw, ,
o 1'



M



L



+ - (u. 2 + v„ 2 + w o 2 )j = M\ i 1 + \ (u x 2 + v x 2 + w^)}







whereas the law of increasing entropy ("More entropy
flows in than out") requires
Mvt^ = M 7q
The phenomenon that results if M f- is called
a shock . The problem then spli ts into two parts, one
Involving only the normal velocity component u, the other
referring to the tangential components and stating that
they go over the shock front unchanged:
(2)



(2')
(2")

(3)



u



MV o , u 1



MV ] _,



Mu Q + p o = Mu x + Pl ,

1 2 _ . 1 2

1 + — U = 1, + rrU, c

2° 121



V



V, , W
1' O



w



1



If, however, M = 0, we obtain a vortex sheet or slip

stream characterized by the relations

u = u, = 0, p = p, :
o 1 ' ^o *±

the fluid in [1] glides tangentially over that In [0] and
has the same pressure on both sides. Our object will be

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the study of shocks in an arbitrary fluid.

After introduction of the values (2) of the
velocities (2') gives

(4) M 2 = Pi - Pq

V o " V l '
and thereupon (2 1 ') yields a relation between the two
the rmo dynamical states Z and Z, ; namely,

h ' i = |m 2 (v o - v x ) . (V Q + v x ) = l( Pl - Po ) (V q + v x ) oi

E l - E o = ^1 + Po> (V o - V

Hence the problem of shocks is reduced to a study of this

relation between two states

H = H(Z r Z o ) = ( El - E ) - |( P] + P )(V o -r Vl ) = C
(Hugoniot equation).

' e are only interested in shocks in one and two
dimensions. If in the latter case the normal unit-vector
of the shock front In the direction [0] — S* [1] is (■:* (3 )
instead of (1, 0) the formulas (2'), (3) for the velocities
read as follows:

(5) c/.u o + /3v q = MV Q , ocu x +/3 Vl = MV 1 ,
(5') -/3u Q + ocv Q = - /^u x + Xv 1 .

A simple analysis' (which will presently occupy
us in much greater detail) reveals that the value of M
resulting from (4) is of the order i,q where q is the
acoustic velocity. Thus we are dealing with a situation
like this: A certain quantity might be zero, but if it
is not, it is even 2s 1. Clearly quite different circum-

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stances must be responsible f»r the two phenomena of
shock find slip stream and M = is in no way to be con-
sidered here the limiting case of a non-vanishing M.
The problem of the shock layer will be compared to that of
the slip layer in §9 by taking viscosity and heat conduct-
ivity into account and then ]e tt?ng them tend to zero. In
particular, this passage to the limit will explain why a
shock is not conservative with respect to entropy though
it conserves energy.-

2. Thermodyncmical assumption s

Next we specify our assumptions concerning the
thermodynamical behavior of our fluid. They will be enumer-
ated, I-IV.

I , I nfinitesimal adiabatic increase of pressure effects
compression :

(g)- < °-

II-, The rate of compression - -r— diminishes in thi s



proc ess :•*



d2v > 0.
dp2/ ad



These local hypotheses will be supplemented by two assumptions

of "global" character.

III. In the continuous pr ocess o f adiabatic compression

one can raise pressure arbitr a rily high..



•; it can be made plausible that condition
II is essential for the formation
of a shock wave*



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,-f - •*



. ..-. ._.-* ,



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IV , The state Z is uniq uely specified by pressure and
specific volume, an d the point s (p ,V) representing the
p ossi ble sta tes Z in a (p, V ) - diagram form a convex regi on..

It would be more natural but less elementary to
divide Iv into a local and a global part, the local postu-
late asserting that in the neighborhood of a given state Z
the variables p,V can be used as parameters for the speci-
fication of states; in other words that the projection
Z -> (p,V) of the manifold j/ of states Z upon a (p,V) -plane
is locally one-to-cne. The global part would assert that
the projection of 4 covers a convex region / in the (p,V)-
plane and that Z being any given state and (p , V ) its
projection, one never runs against an obstacle when, start-
ing at Z , one lets Z vary so that its projection ( ,\l) follows
a given pa.th in L beginning at (p ,V ) ("continuation").-
All this means that K is a covering surface over the



convex X without signularities and relative boundaries.



But the convex /(p,v). The adiabatic process
of compression Is thus defined by

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-I i



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^dp + ^dV =

and since -*-2, ~£ cannot vanish simultaneously, eq. (29),

V.) 7) I

hypothesis I requires that ^r 1 -, ^rh are of the same sign:

> p ' >V 7) p ' 3 V

As the region 4 is connected, the one set or the other will
hold every whe re . We assume the first:






> Q, %-l > 0.



(The other alternative would make little difference.)

Infinitesimal adiabatic compression is now described by

dp = a*,dt, dV = b#.dt; a-::- = M, b-::- = - 1~

(a posi tive dt corresponding to an increase of pressure) .

Hypothesis II yields

d / b' ,N \ ^ « n „ v.-»-da'" -::- db""" . ~

"XT I ~ - ) > 9 or b -rx -a ■ 3 - c < 3,

Qt^ a v/ d t d t '

1 - e * b ( r^ a + Vrrb / - a —-a - - + ^— b" / =

*2db* *.#/3a* >b*^ . h #23 a*
(lit) ^ /£» XS _ ^ ^ ^ ^ ^, 2 < c<

^ p 2(^) 3p^v ? vJp av«^p/

In the future we shall use hypotheses I and 11 in their
analytic form (I') and (II 1 ).

Our next move consists in developing a number
of consequences from the assumptions I-III. Let (a,b) T (0,0)
be two given numbers which determine the direction of a
straight line through Z = (p , V ) ,

p = p o + at, V = V c + bt,
the half-line or ray being obtained by the restriction t ^>

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of the parameter t. We follow the straight line or the
ray as long as it stays within -Y, end form

dt 3p ^V ^p^ Sp^V 3v^
According to ( I ' ) , "7 is positive (negative) and hence ^l in-
creasing (decreasing) along the whole straight line, provided
a 2? 0, b .^.0 (ago, bgO).

Lemma 1 . If ^ =0 for a certain vr lue of t, then

d7 < for the same value ,
d~t~

"roof.. Substitute the values of a and b derived

from 7 = into -ppp- and use the inequality (II 1 ).

Lemma 2. If 7' < for t = C, the n l'< for
t > 0.

Proof. First assume 7 < for t = 0. Should
7 change sign as Z travels along the ray, it would have to
vanish somewhere. Suppose this occurs for the first time
at t = t, . As 7 < C before t reaches this value, 7 passes
the zero level at t, ascending; hence -T-r- ^ for t = t, .
But these two relations 1-0, -rrf- 2: for t = t, ,
contradict Lemma 1.

If y \ - for t = we have ^f- < for t = 0;
consequently 1 / becomes negative immediately after the point
Z has started on its way, and from then on, as we have seen,
'I' must remain negative.

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1 o turn to III. Starting at a given po^'nt
Z = (P > V ) and raising p conti nuali y, we follow the adi-
abatic through Z ; we thus obtain a continuous monotonely
descending function V(p) , and according to III we can make
p travel over the entire infinite interval p ^> p . -Vhile

this happens, the directional coefficient s = ^ " P° of the

[email protected] - V
straight line joing (p o ,V ) with the point (p,V)' on the

adiabatic Tf = °7 increases monotonely from a certain positive

value m to + p of the adiabatic
on the ray from (p , V ) of a given direction s, provided
s lies between m and-o^.

The analytic proof for these intuitively evident .
statements runs as follows. Form, the (non-total) differential

dr = (V Q - V)dp + (p - p Q )dV,
which vanishes along any ray through Z . ' e want to show
that along the adiabatic dr is positive for a positive incre-
ment dp;

r = (v - v) + ( P - p o )g > e.

By hypothesis II

< 6 > f = z o 1

linked by Hugoniot's equation H = the follo wing inequal:; ties

hold.

(12) N* = (p 3 - P Q ) (V Q - V x ) > 0,

(15) (p 3 - p Q ) + m o (V 1 - V Q ) > r (p 2 - p Q ) + m 1 (V 1 - V Q ) <

Before proving the theorem let us discuss its
physical significance * We have seen that elimination of
the velocities from the conditions for a shock lead to
the Hugoniot equatio n and the inequality N" ^ 0. V 1 e now
real ize that wo may omit the supplement.-. ry relation N" >
because it follows, even in the sharper form >t" > 0, from
the Hugoniot equation.

The adiabatic derivative

dg = _ v 2 # d£ = 2;
df dV

is the square of the "acoustic velocity" q. Assume V > V, .

Ther. the two relations (13) give

(14) m < M 2 = P l " p o < *

V o " V l 1
or P q < |?;| < P^q , thus confirming a statement made in

£1 on the magnitude of M. In terms of the velocities u Q ,u 1

given by

2 Pi " P Q „2 2 _ P l p o „2
c V Q - V 1 o 1 V - V 1 1

(14) may be written as

(15) !u o l > q Q , lu 1 !< q 1 ?

re lat-vul y to the shock front the flow in [0] ; | S_ supersonic,

in [1] subsonic.

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Proceeding to the nroof of our theorem, let us
travel from Z along that ray which passes through Z-, and
hence set in the two lemmas

a = p x - p Q , b = V;L - V Q .
Were a ^ 0, b ^ 0, then V would be positive and hence 7
monotone increasing along the segment Z Z, , which contra-
dicts the relation (11). Consequently this combination as
well as the other a < 0, b < are ruled out, and therefore
ab < 0, or (12) must hold.

Next consider

1 3p sv
Were Y g for t = 0, thon "*J v 'oulc be negative and hence
*J would monotonely decrease wh'le Z travels along the
straight segment from Z to Z-, , in contradiction to the
equation (11). Therefore

a£T) + b (||) > 0,
^P/O \3V/o

and this is identical with the first of the inequalities
(13), from which the second follows by interchanging: Z Q

ana Z, .

But our argument shows much more, while Z travels
along the ray from Z q passing through Z-j_, °] starts with
a positive value; because of (11) it must change sign be-
fore Z reaches Z, . Put ')' remains negative from the moment
it vanishes for the first time. Rise and fall of H = H(ZZ Q )
along the ray is coupled with that of 01 by the relation

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dH = Tdw . Hence H first rises monotoncly to a positive
mr.x ; mum ana then decreases, on the descent pas si ng through
for Z = Z-,. This description shows thr.t His positive from
Z to Z-, and negative after. In particular:

On the ray from Z through Z, the point Z-. is
the only one which satisfies the Hugoniot equation H = C.

According to (12) either the inequalities
P x > p Q , V Q > V^ or p Q > ■p 1 , V. > V Q hold for two distinct
states Z , Z-> , satisfying the Hugoniot equation H(Z-,, Z ) = f ,
i: e shall indicate these altern; tives by Z, > Z , Z, < Z

respectively. By the Hugoniot contour #y(for a given Z )
we understand the locus of all po'nts Z-, = (p^V-,) for which



1 v ^l u l



H(Z,Zj = and Z, > Z .

1 o 1 o



(It is quite essential for our argument to pick this upper_
: 1 > Z o'



branch Z-, > ! „.) Our above result may then be stated thus:



P-, " P n
On the Hugoniot contour s = — u lies between m andO° t

v o - v x

s is a uniformizing parameter inasmuch as the value of s
specifies uniquely the point Z-, . But it must be borne in
mind that so f ; r we have not yet proved that to every pro-
assigned value of s > m there actually corresponds a point
on the Hugoniot contour; we only know thr.t there cannot be
more than one .

4._ Entropy ana p; ra mi tr ' za tion of the H ugo niot l 'ne

For a moment let us return to the upper branch ^V

of the adiabatic through Z , on which p increases monotonely

from p to +o^ and the directional parameter s = ^ ~ j ? T fram m

o

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,&



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too». Moving along it we have by (9) dH = -gdr = - in. dp
.'.•rid hence (7) implies

Lemma 5. H = H(ZZ ) is negative along Ot .

As r. matter of fact, we even know that H ?' s

falling, R > 0, and falling with increasing rapidity,
~ > 0, eq. (6). The explicit formula giving H(Z°Z )
for any po^'nt Z on Ly is

H ( z %) = " \ f P Vp - - \ A P °(p - p Hp°- p^p < o.

7 Po ^'P
This lemma is instrumental in establishing the

following two proposi t ionsr

Theo rem 2. For rn y two dist i nct s tates Z , Z,

atlof;:'n g the Hugoniot equ ati on one has "i > V or fl < J

a ccord i ng to whether Z-, > Z or Z, < Z »

Theorem 5 . The H u goniot co ntour is a simple line
starting from Z o n which s and V are monotone increasing
(s traveling from m to +.

Physically this means that when the unit quantum
of fluid in the state (p , V ) is sealed up in a vessel
of volume V and then her ted, its pressure can be raised
to •' rbi trarily high values. I na:"nte: : n:

Theorem 4. Under t he addition a l hypothesis V
t he function p( s) representing pressure as function o f s
alo ng the Hugoniot line, tends to infi nity with s — > o* 7 .

p tends to infinity along the adiabatic line
(hypothesis Til) as well as 'long the vertical V = V

(hypothesis v) ; the same behavior along the Hugoniot line
will be shown to be r consequence of these two circumstan ces ,
We a r gu e as f o 1 1 o w s .

Let h be an arbi trarily large positive constant.
The section p ^ p ;g p + h of the vertical v = V lies
in \ and hence the same is true for a cert." in strip 3 of
small width a ,

V Q -S g ViV , P SpgP + h
to the> left of it. may ascertain two positive constants

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c, C such that ~- 5; c, ^ g C in S, Consider the section

g t g h of the ray p = p Q + t, V = V Q - 7l t . If the

positive cons tant A- which determines its direction is

r

less than ^— that section lies in S and therefore

h

dt Tp A "^v > °

provided A< c/G. Previously we have characterized the
direction of such a ray by the number s = !//[_ . Hence
^ is on the ascent along the whole segment p < p g p + h


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Online LibraryHerman WeylShock waves in arbitrary fluids; a note submitted to the Applied Mathematics Panel, NDRC → online text (page 1 of 3)