Courant M athematics and

Computing Laboratory

U . S. Depa rtmen t ofEne rgy

Lectures on Combustion Theory

Edited by

SamuelZ. B urstein ,Pete r D . Lax,

and G a ry A . Sod

Resea rch and Developmen t Repo rt

a red under Con t ract EY-76-C-02-3077

w it the O ffice ofEne rgy Resea rch

M athematics and ComputingSeptembe r 1 978

New York University

UNCLASSIFIE D

C ourant Mathemat i c s and C omput ing Lab o rato ry

New Yo rk Unive rs i ty

Mathemat i c s and C omput ing COO-3 077 - 1 53

Le c ture s given in a Sem inar he ld during Spring

s eme s te r 1 977 at the C ourant Ins t i tute

E d i te d by

Samue l Z . Burs t e in , Pe te r D . Lax , and Gary A . Sod

Se p tembe r 1 97 8

U . S . De partmen t o f E ne rgy

C ont rac t E Y - 76 - C- oa- 3 o77

UNC LASS IFIE D

Tab le of C ontent s

The Nume ri c al S olut i on of the E quat i on s of Flui dDynami c s

by Pe t e r D . Lax

Le c ture 2 . On the Mathemat i c al The ory of De f lagrat i on s andDe t onat i ons

by K . O . Fr ie dr i chs

Chemi c al Kine t i c sby Pe te r D . Lax

Random Cho i c e Me thod s wi th Appli c at i ons t oRe ac t ing Gas Flow

by Alexandre Joel Cho rin

A Nume ri c al S tudy of Cyl indri c al Implo s i onby Gary A . Sod

C ombus t i on Ins t ab i li tyby Samue l Z . Burs t e in

The ory of Flame Spre ad Ab ove Sol id sby Wi lli am A . S i r ignano

One - Dimens i onal Analys i s of C ombus t i on in aSpark- Igni t i on E ngine

by Wi ll iam A . S i rignano

The Mas s Burning Rate of S ingle C oal Part i c le sby Irvin Glas sman

S tudie s of Hyd roc arb on Oxi dat i on in a Fl owRe ac t o r

by Irvin Glas sman , Fre d L . Drye r , andR . C ohen 2 8 2

Le c ture l l . Some Pe rc e pt i ons on C ondens e d Phas e F lameSp re ad ing and Mas s Burning

by I rv in G las sman

Int roduc t i on

Thi s volume c ont ain s the e lab orat i ons of le c ture s at a

sem inar he ld at the C ourant Ins ti tute in the s p ring of 1 977 on the

mathemat i c al aspe c t s of c ombus t i on . The purpo s e of the seminar

wa s t o put the achi evement s and pr ob lem s of c ombus t i on the ory int o

sharp fo c us and t o b ring them t o the at tent i on of the mathemat i c al

c ommun i ty , in the hope that , j us t as in the pas t , mathemat i c al

me thod s wi ll she d light on the s e the orie s , and that mathemat i c al

i de as wi ll l e ad t o new and e ffi c i ent c omput at i onal proc e dure s .

The fi r s t half of the s eme s t e r was devote d t o s ub j e c t s that

we re re as onab ly we ll unde r s t ood as mathemat i c s ; the speake r s we re

mathemat i c i ans . Af te r the s p ring re c e s s the s eminar wa s devo te d

t o s ub j e c t s not ye t mathemat i c ally d ige s t e d ; among the s pe ake r s

the re we re engine e r s , chemi s t s , and phy s i c i s t s wi th sympathy in

the i r he art s for mathemat i c s .

The fi r s t part s t ar t s wi th a pape r by Pe te r D . Lax whi ch is

a review of tho s e nume ri c al me thod s in flui d dynami c s that are

e s pe c i ally pr om i s ing for re ac ting flows . Thi s is followed by a

re port p re pare d by K . O . Fri e d ri chs f or the Navy in 1 946 ,

ed i te d and p re s ente d by Gary A . Sod . The thi rd pape r by Pe te r D .

Lax i s a b ri e f int roduc t i on t o chem i c al kine t i c s , fol lowe d by

pape r s by Alexand re Cho rin , o f the Unive r s i ty of C al i fo rni a at

Be rke ley , on re ac t ive f low s , by Gary A . Sod on a fi rs t s te p t o

mode ling flows in an eng ine,and by Samue l Z. Burs t e in on c ombus

t i on ins tab i l i ty .

The s e c ond par t c ons i s t s of two pape r s by Wi lli am S i rignano

of Prince t on Unive r s i ty on flame sp re ad and re ac t ive flows in a

one - d imens i onal engine , foll owe d by three pape r s by Irvin Glas sman

of Pr ince t on Unive r s i ty on the burning rate o f S ingle c oal part i

c le s , on hyd roc arb on oxi dat i on in a flow re ac t or , and on f lame

spre ad ing .

We di d not inc lude in thi s c ol le c t i on an int e re s t ing le c ture

by Loui s Howard of M . I . T . on hi s work with Nancy Kope ll on re ac t i on

d i ffus i on e quat i ons , the c ontent of whi c h i s c on taine d in the s e two

pape r s

N . Kope l l and L . N . Howard , Plane Wave Solut i on s t o Re ac t i on

L . N . Howard and N . Kope l l ,

" S lowly Vary ing Wave s and Shock

Math . , 56 , 95 ( 1 977 )

Anothe r inte re s t ing le c ture not inc lude d wa s by Jame s Muc ke rman o f

the Brookhaven Nat i onal Lab o rat ory of the Department of E ne rgy on

the c alculat i on of b im ole cular rat e c on s t ant s b as e d on the thre e

pape r s

J . Mucke rman and M D . Fai s t ,

" Ra te C on s t ant s fr om Monte C ar loQua s ic la s s ic a l Tra j e c t ory C al culat i ons : The Us e of Impor tantSampl ing s ,

" t o appe ar .

P . A . Whi t lo ck , J . Mucke rman , and E . R . Fi s he r , The ore t i c alInve s t igat i on o f the E ne rge t i c s and Dynami c s of Reac t i ons ofO ( d ) with H

2,

" s ubmi t t e d t o J . Chem . Phy s .

" The ore t i c alInve s t igat i on o f the E ne rge t i c s and Dynam i c s of Re ac t i ons ofo( 3 p d ) wi th H

2and C ( d ) wi th H

2 ,

" RIE S Te chni c al Repo rt ,

Wayne S tate Unive rs i ty , De t r oi t , Mi chigan

We he art i ly thank all o f the s pe ake rs for the i r par t i c ipat i on .

Our thanks are due t o Ac adem i c Pre s s , C amb ridge Unive rs i ty Pre s s,

Gordon and Bre ach Sc ience Pub li s he r s , and Pe rgamon Pre s s for

pe rm i s s i on grante d t o re print the art i c le s of Profe s s or s Chor in ,

Glas sman , Si r ignano , and Sod .

Samue l 2 . Burs te in , Pe te r D . Lax , Gary A . Sod

C ourant Ins t i tute , July 1 97 8

vi i

The Nume ri c al S olut i on of the E quat i ons o f Flui d Dynami c s

Pe te r D . Lax

C ourant Ins t i t ut e of Mathemat i c al Sc ienc e sNew Y ork Unive r s i ty

1 . Int roduc t i onThe e quat i on s o f flui d dynami c sThe o ry o f sho ck wave sThe me thod o f frac t i onal s t e p sDi ffe renc e approximat i on o f c ons e rvat i on lawsThe me thod s of Godunov and Glimm

E nt ropy and vi s c o s i tyN

am-EN

D.

)

1 . Int r oduc t i on

The the ory o f Chapman and Jugue t for de t onat i ons and

de flagrat i ons de s c r ibe s re ac t ing flow s in the limi t as the re

ac t i on rate goe s t o 0 0 , vi s c o s i ty , d i ffus ivi ty and he at c onduc

t i on go t o O . In thi s the ory the t rans i t i on from burnt t o unburnt

ga s take s pl ac e ins t ant ane ous ly ove r an infini t e ly thin re ac t i on

z one . In many pr ob lem s o f c ombus t i on one is int e re s te d in a fine r

re s olut i on o f the re ac t i on z one ; thi s is po s s ib le only by s olving

flow e quat i ons whi ch c ont ain an ade quate de s c r ipt i on of all re le

vant chemi c al and phy s i c al p roc e s s e s : the rate s at whi ch the

re ac t i ons p roc e e d , the c onve rs i on o f chemi c al ene rgy t o he at , the

c onduc t i on o f he at , the d i ffus i on o f the vari ous s pe c i e s , and the

e ffe c t of vi s c ous forc e s . In the t rad i t i onal engine e ring l i te ra

ture s uch pr ob lem s are t re ate d analy t i c ally , a t the c o s t of

dras ti c s impl i fi c at i on which s t i ll re tain sh red s of the phy s i c o

chemi c al proc e s s e s re s pons ib le fo r the phenomenon under inve s t iga

t i on . Thi s i s the only avenue open,un le s s

,as P . A . Wi l li am s

remark s in his t re at i s e on c ombus t i on the ory,

" one i s wi l ling t o

expend the lab or re qui re d t o ob t ain c omp le t e nume ri c al s olut i on s

Wi th the advent of mode rn c ompute rs and mode rn nume ri c al me thod s

for c alculat ing flui d flows , s c i ent i s t s are Wi ll ing and ab le

t o expend s uch lab or , although the c omple t e mode lling o f a c omb us

t i on pr ob lem , t rac ing do z ens of int e rme di ary pr oduc t s part i c ipat ing

in the re ac t i on , i s b eyond the s c ope of pre s ent day nume r i c al

me thod s . Much current re s e arch i s d i re c te d at adopt ing exi s t ing

me thod s for c alc ulat ing flui d flows t o the c alc ulat i on of re ac t ing

flows . Many of the s e exi s t ing me thod s employ s t ab i l i z ing devi c e s ,

of whi ch art i fi c i al vi s c os i ty i s the m os t prev a l ent , tha t c aus e

only marginal and ac c e pt ab le nume ri c al inac curac i e s f or non

re ac t ing flows but would d i s t o rt e s s ent i al feature s of re ac t ing

flows such as f lame ve loc i ty that de pend on a b alanc e b e twe en

t rans port,he at c onduc t i on , and ene rgy p r oduc t i on . A . Chor in has

made the impo rt ant ob s e rvat i on that among the many avai l ab le

me thod s the one deve lope d by G limm, i s the fre e s t of a r t ifi

c ia l encumb rance s . In Se c t i on 6 o f thi s le c ture we de s c r ibe how

and why Gl imm ‘s me thod wo rk s fo r nonre a c t ing fl ows ; the adopt i on

t o re ac t ing flows i s de s c rib e d in a sub s e quent le c ture by Chorin .

Anothe r phenomenon of nume r i c al s cheme s whi ch are of highe r

than minimal orde r of ac c uracy o s c i l lat ory b ehavi or ne ar a

d i s c ont inui ty,whi ch re s ult s in an ove r shoot ing of pe ak value s .

Thi s i s p re s ent in Lax-Wend roff type di ffe renc e s cheme s , and als o

in s pe c t ral s cheme s , on ac c ount of the Gibb s phenomenon . In

ord inary flui d dynami c s such an ove r shoot , when not exc e s s ive, is

me re ly an ae s the t i c b lemi sh ; in re ac ting flows , whe re the rate of

chemi c al re ac t i on is s o s ens i t ive t o t empe rature , an ove r shoot

would premature ly t rigge r off igni t i on and would fals i fy the t ime

hi s t o ry of burning . One cure f or ove rshoot ing in LW or o the r

type s of highe r orde r s cheme s l i e s in hyb ri di z at ion c omb ine d wi th

art i f ic i al c ompre s s i on , as deve l ope d by A . Harten , The c as e

for s pe c t ral me thod s l i e s in apply ing s ome summat i on me thod .

A c ons e rvat i on law as s e r t s that the change in the amoun t of

a sub s t anc e c ontaine d in any po rt i on of s pac e i s due t o the flux

of that s ub s t anc e ac ros s the b oundary of the po rt i on of s pac e unde r

c ons i de rat i on . Le t 's denote the dens i ty of the sub s t anc e by u , i t s

f lux by f , and the port i on of s pac e unde r c on s ide r at i on by G .

Then the c ons e rvat i on l aw s ay s that

t t

udx f ° v d t O

G s 5 6C

whe re v denote s the outward normal t o the b oundary 8G of G . Us ing

the d ive rgenc e the orem the b oundary inte gral te rm c an be wri t t en as

d iv f dxd t

5 G

Le t t ing s tend t o t and G shrink to a point we d educ e that at eve ry

point whe re u and f are d i ffe re nt i able , the di ffe rent i a l c onse rva

ti on law

U d iv f o

is s at i s fie d .

The re s t of thi s s e c t i on i s devo te d t o a b ri e f d i s c us s i on

the c las s i c al c ons e rvat i on law s of mas s , momentum and ene rgy in

fluid dynam i c s . We shal l show how vari ous t ran sport me chani sm s c an

be expre s s e d by a sui t ab le c hoi c e of f lux .

We shall us e the not at i on

p mas s dens i ty

M momentum dens i ty

E ene rgy dens i ty

Momentum and ene rgy den s i t i e s c an b e exp re s s e d as f ol low s

M pV E pe-t %»

pV2

whe re

<1 II flow ve loc i ty

e inte rnal ene rgy pe r uni t mas s

S inc e the f lui d i s s t re aming pas t G wi th ve lo c i ty V , mas s i s

c onve c t e d out of G at the rate

pV’V dS

due t o thi s c onve c t i on i s

fc onv mas s c onv

flux of mas s i s due t o d i ffus i on ; thi s pr op or

negat ive gradi ent of mas s dens i ty

( 2 ’ 4 ) di ff fmas s d i ff" D gr a d P

D is c alle d the c oe ffi c i ent of d i f fus i on .

Momentum flux is the sum of two kind s of t e rm,e ach repre

s ent ing two d i s t inc t t rans por t me chani sm s : c onve c t i on,and impul s e

of f orc e s exe rt e d by the flui d . The c onve c t ive te rm is ent i re ly

analogous t o ( 2 . 4 ) c onvand i s of the form

1 i( 2 ' 5 ) c onv fmom c onv

M V

whe re the s upe r s c r ipt 1 re fe r s t o the ith

C ar te s i an c omp onent ,

i To de rive the impul s ive te rm s , denot e by F the fo rc e

pe r uni t are a exe rt e d by the flui d ac r o s s a surface e lement through

a point x and wi th outward normal v . It i s a b as i c law o f

c ont inuum me c hani c s that F has the form

F Pv

whe re P, c alle d the pre s s ure tens or , i s a symme t r i c mat rix func t i on

of x . The impul s e of the for ce F of f orm c hange s the

m omentum o f the flui d c ontaine d in G at the rate

PvdS

The rate at whi ch momentum in the i c oordinate di re c t i on change s

i s thus

f.i

momentum P

Fo r a nonvi s c ous f lui d the p re s s ure t ens or i s s c alar , i e .

the i dent i ty mul t ip li e d by the s c alar pre s s ure p

P DI

In thi s c as e the flux of the momentum in the 1 d i re c t i on i s

i i( 2 . 7 )

pfmomentum p

uni t ve c t or in it h

d i re c t i on

For vi s c ous flui d s the s c alar p re s s ure t ens or ha s t o be augmente d

by a mat rix whi ch for inc ompre s s ib le f lui ds t ake s the s imp le fo rm

u BVl

OVJPvis c e (at; s

q

)

u i s c al le d the c oe ffi c i ent of vi s c o s i ty . The flux of momentum in

the ith

d i re c t i on due t o v i s c ous fo r ce s c an be ob tained by s e t t ing

int o

i

v i s c m omentum vi s cu i

V -+ VX1

E ne rgy flux i s the su m o f thre e t e rm s ; the fi r s t re pre s ent s

the e f fe c t of c onve c t i on and ha s the value

f E Vc onv ene rgy c onv

The work done by a fo rc e of form change s the ene rgy c ontained

in G at the rat e

V ° FdS Vo ds

https://www.forgottenbooks.com/join

d d t

and thi s has t o e qual the le ft s i de of thus we ob t ain the

e quat i on

dx f ° vd t d dt

G s 3 5C

From thi s we c an de rive as b e f ore the analogue o f the d i ffe rent i al

form whi ch for e qua t i on i s

ut

-t div f S

The re are many pr oc e s se s in whi ch s ourc e t e rm s play an import ant

r ole , par exc e llence chemi c al ly reac t ing flows ; the c hem i c al re

ac t i on i s a s ourc e of mas s f or the produc t s of the re ac t i on , and a

s ink fo r the re a c t ant s . I f the re ac t i on i s exothe rmi c i t is a

s our ce of heat ene rgy , i f end othe rmi c a s ink of ene rgy . The s e

s our ce s t rengths wi ll b e den ote d a s foll ows :

mas s i

whe re i lab e l s the s pe c i e s par t i c ipat ing in the chemi c al r eac t i on

and ri, c alle d a re ac t i on rate , is the rate at whi ch the dens i ty

of the s pe c i e s i i s changing as a re s ult of the chemi c al re ac t i on .

The re ac t i on rate s ri

at x de pend on the de ns i t ie s of the s p e c i e s

par t i c i pat ing in the re ac t i on at x . The s t rength of the heat

s ourc e is denote d by

ene rgy

whe re q i s the rate of ene rgy re le as e during the re ac t i on ; q is a

func t i on of the dens i t ie s of the re ac t ant s .

Anothe r ene rgy t rans fe r me c hani sm that is de s c rib e d in te rm s

of s our ce s and s ink s i s rad i at i on ; thi s me c hani sm be c ome s impor

t ant at high tempe rature s .

The fo re going d i s cus s i on shows that the c ons e rvat i on laws of

f lui d dynami c s may c ont ain a varie ty of f luxe s and s ourc e te rm s .

In Se c t i on 4 we shal l s how how the me thod o f frac t i onal s t ep s c an

be us ed t o di s ent angle the vari ous t rans fe r me chani sm s f rom each

othe r in the c ons t ruc t ion of appr oximate nume ri c al s oluti on s .

It follows f r om the d i s cus s i on in Se c t i on 2 that i f diffus i on ,

vi s c os i ty , heat c onduc t i on and chem i c al re ac t i ons are negle c t e d ,

the law s o f c ons e rvat i on of mas s , momentum and ene rgy are

pt+ diV M = O

it+ div ( M V ) -t pM II 0

X .

1

Et

-t div ( EV + pv) 0

To make thi s sy s tem s e l f- c ont aine d we have t o ad j o in re lat i ons

( 2

( 3 2 ) M pV E pe +27 p

V

and the e quat ion of s tate , whi ch re late s thre e the rmodyn ami c

vari ab le s , say e , p , p

1 — 1 0

( 3 -3 ) P = P ( e i p )

The s imple s t e quat i on of s tat e i s the c las s i c al po ly t r op ic re lat i on

( 3 p ( y— l ) ep y c ons t .

The sy s t em of c ons e rvat i on laws s upplemente d by and

i s of the gene ral fo rm

u§ -+ div fk

O k

whe re e ach fk

i s a fun c t i on of the dens i t ie s them s e lve s

k k l N( 3 -5 ) f = f ( u ,

To s imp l i fy the di s cu s s i on we turn now t o one - dimens i onal flows ;

a b rie f di s c us s i on of the two- d imens i onal c as e wi l l b e given at the

end . Dropp ing dependenc e on y and z in le ave s

uk+ f

k= 0

To analyz e s olut i on s of such a sy s t em we c ar ry out the di ffe rent i a

t i on wi th re s pe c t t o x in we ge t a fi r s t orde r sy s t em whi c h

in mat rix no tat i on c an be wr i t t en as

ut

-t A ( u ) uX

O

l N twhe re u ( u and A i s the mat rix gr adi ent of

f wi th re s pe c t t o u

kBf

( 3 -8 ) A ( akj) a

kj‘

5 53

Thi s i s a sy s tem of quas i line ar e quat i on s ; i t i s we l l kn own that

in orde r for the ini t i al value pr ob lem t o b e prope r ly p os e d,

value s a them s e lve s func t i ons of U . The e igenvalue s have

a ve ry int e re s t ing phy s i c al int e rpre t at i on : they are the loc al

de s c r ibe d by e quat i ons The s e are c alle d the charac te ri s t i c

ve loc i t ie s of the e quat i on

The one - d imens i onal ve r s i on o f the sy s tem

i s hype rb oli c in thi s s ens e , pr ovi de d that p re s s ure i s an

P

57, when p i s given a s func t i on of p and e

dp entropy=cons t

it s de rivat ive wi th re s pe c t t o p at c ons t ant ent ropy is given by

g-a

§ +

Pent ropy=c ons t

The inc re as ing charac te r is expre s s e d by s e t t ing

ent ropy=cons t

The quant i ty c c as e

we have

2c

P

The charac t e ri s t i c ve l oc i t ie s fo r the e quat i on dynami c s are

If the e igenvalue s are d i s t inc t , thi s c ond i t i on s uffi c e s , in c aseo f a mult iple e ige nvalue add i t i onal c ond i t i on s have t o be imposed .

l l

1 - 1 2

V- c C V+ c

Ac c or ding t o the gene ral the ory of nonl inear hype rb oli c e qua

t i ons of fo rm we may pre s c r ibe the ini t i al value s u ( x , o)

arb i t rari ly ; the c orre s pond ing s olut i on U at the p oint x , t is

uni que ly de te rmine d by the ini t i al dat a , in fac t by the ini t i al

po int ( x , t ) the domain of de pendenc e i s the smalle s t int e rval

c ontaining the inte r s e c t i on of all charac te r i s t i c curve s thr ough

x , t wi th the ini t i al l ine t O ; a charac t e r i s t i c c urve is one

whi ch propagate s wi th one of the charac te r i s t i c ve l oc i t i e s , i . e .

s at i s fi e s one of the d i ffe ren t i al e quat i ons

E ?a u ) u u ( x , t ) k

In ge ne ral s olut ions o f a nonline ar hype rb ol i c e quat i on

deve lop s ingulari t i e s afte r a c e r tain t ime has e laps ed . The s ourc e

of thi s b re akd own i s e as i e s t s e en for a s ing le c ons e rvat i on law

ut+ f ( u )

xo

whi ch c an b e wri t t en as

( 3 -1 4 ) ut+ a ( u ) u

xO a _ '

du

The le ft s i de of c an be inte rpre te d a s the de rivat ive of u

in the charac t e r i s t i c di re c t i on

Eo , whe re

H= a ( u )

Thi s e quat i on s ays that u i s c ons t ant along the charac t e r i s t i c

1 2

c urve , but then the s pe e d a ( u ) i s c ons t ant along the cha ra c te ris

t i c curve , and s o i t foll ows that the c harac te r i s t i c curve i s a

s t raight line . Now le t xl

and x2

denot e any two point s on the

ini t i al l ine t O , u ( x , o ) the p re s c r ib e d ini t i al func t i on . The

s pe ed s of the charac t e ri s t i c line s i s s uing fr om the s e p oin t s are

al

and a2

i f the s e two line s are on c ol li

s i on c our s e , then at the point ( x , t ) o f the i r inte r s e c t i on u ( x , t )

ha s t o b e e qual t o b oth u and u whi ch shows that no s olut i on1 2

’

c an exi s t bey ond that t ime , at leas t not a c ont inuous one .

What doe s exi s t bey ond that t ime ? E xpe r iment s di s c los e the

appearanc e of di s c ont inuous s olut i ons . In what follows we out line

b rie fly the the o ry of the s e . For a mathemat i c al de f ini t i on of

di s c ont inuous s olut i ons we have t o go b ack t o the phys i c al de fini

t i on of a c ons e rvat i on law : we s ay that u i s a s olut i on of the

sy s t em of c ons e rvat i on law s i f the inte g ral re lat i ons

are sat i s fi e d for al l d omains D and all t ime s t and s . An ent i re ly

e quivalent formulat i on is t o re qui re that the e quat i ons be

mult i ply by any smooth te s t func t i on w ( x , t ) that vani she s

for lx l large , and i f we in tegrate by part s , the re sult ing int egral

re lat i on

( w uk

-r gra d u . fk) dxd tt

0

hold s for k

Suppo se that u i s a p i e c ewi s e smooth func t i on in reg i ons

s e parate d by smooth s urface s . It i s not hard t o show that u s at i s

fl e S in the d i s t ribut i on s ens e i f f

i ) u s at i s fi e s e quat i on p ointwi s e in eac h smo oth

regi on ,

i i ) ac ro s s e ach s urfac e of di s c ont inui ty the j ump re lat i ons

( 3 -1 7 ) -v

ho ld , whe re deno te s the jump of the quant i ty in b racke t s

ac r o s s the s urface of di s c ont inuity , v the d i re c t i on in whi c h the

s urfac e pr opagate s , and s the s pe e d wi th whi ch i t prop agate s .

The s e re lat i ons , c alle d the Rankine — Hug oni ot c ondi t i on s and

abb revi ate d a s R— H , are e as i ly de rive d fr om or fr om the

more phy s i c al e quat i on

In c a se of one s pac e dimens i on the j ump re lat i ons t ake the

form

s iuk] [ f

k]

whe re s i s the s pe e d with whi ch the d i s c ont inui ty pr opagat e s fr om

le ft t o r ight .

Fo r small di s c ont inui t i e s the jump re lat i on has a s imple

c ons e quence . Us ing re lat i on we c an wr i t e

[ f ] A [ ul

whe re A is ve ry c lo s e t o A ( u ) gr a duf‘

. Sub s t i tut ing the ab ove

int o g ive s

1 4

https://www.forgottenbooks.com/join

o fo r x/t

x , t ) for x/tu2(

1 for x/t

He re thre e c ons t ant s t at e s u O , 1 are f i t t ed t oge the r along

two l ine s , whos e pr opagat i on s pee d i s cho s en as s and s

s o that the R- H re lat i on i s s at i s fi e d at b oth d i s c ont inui

t i e s ; thus u2

s at i s fi e s in the di s t r ibut i on s ens e . Finally

we de fine

0 for x/t O

u

3( x , t ) x/t f or O x/t l

l f or l x/t

He re the two c ons t ant s t at e s u O and u l are j oine d c ont inu

ous ly by the func t i on x/t ; thi s func t i on s at i s fi e s e quat i on

pointwi s e .

The thre e func t i ons ul, u

2, u

3have the s ame ini t i al value

u O for x O , u l for l x . C le ar ly many mo re are po s s ib le .

We ab s t rac t f r om the s e examp le s a pr inc iple that e lim inat e s al l

but one of the s e s o lut i ons .

Le t u and u be two c ons t ant s denot ing the s t ate on the le ftt r

and on the right of a di s c ont inui ty . Le t

f£

-' fr

u - uri

1 6

then

u ( x , t )

is a di s c ont inuous s olut i on of in the int e gral s ens e . We

be twe en u and ur

s uch thati

ff

— f ( v )

uz

- v

SE

Sr

c as e s pl i t s o lut i on

u for x/t s

i 2

v for 82

x/t sr

u for s t\ r r

x/

als o s ati s fi e s the int eg ral form of the c on s e rvat i on law

and ha s the s ame ini t i al value as the s olut i on u de fined by

We c all a d i s c ont inui ty that c an be s pli t uns tab le . S olut i on s

wi th uns t ab le d i s c ont inui t ie s are re j e c t e d as phy s i c ally not

reali z able . S tab i li ty i s oppo s i t e o f ins t ab i li ty

s pli t , i . e . i f fo r all v be tween uzand u

r

17

s, 3

s,

Us ing the de fini t ions of s and of SE, s

rwe

c an wri te

Sinc e v l i e s b e twe en u and ur, thi s shows that s i s a c onvex

2

c omb inat i on of s and sr; s o i t f ollows f rom that

t

Si

s sr

Again us ing we c an wri te thi s as

fz

- f ( v ) fE

- fr

f ( v ) - fr

( 3u£

- v - ufl

- uT

v ” Ur

Le t t ing v tend t o ug

or ur

re sp e c t ive ly , and re c al ling that

df/du a ( u ) , we de duc e that

az

sar

Thi s expre s s e s the fac t that s ound wave s or i ginat ing on e i the r

s i de of the di s c ont inui ty pr opagate t oward the di s c ont inui ty .

The the o re t i c al s igni f i c anc e of thi s c ondi t i on eme rge s i f we l ook

upon a di s c ont inuous s olut i on as a s olut i on of a mixe d ini t i al

b oundary value p rob lem , the d i s c ont inui ty se rving a s an int e rnal

b oundary . Cond i t i on guarante e s that eve ry c harac t e r i s t i c

d rawn b ackward from e i the r s i de of the d i s c ont inui ty re ache s the

i ni t i al l ine . Thi s show s that the ini t i al dat a de t e rmine uni que ly

the s olut i on on e i the r s i de of the d i s c ont inui ty ; the R- H

1 8

c ondi t i on then s e rve s as an ord inary di ffe rent i al e quat i on

for the de te rm inat i on of the line o f d i s c ont inuity x x ( t ) , wi th

s dx/dt . We shall c all the charac te r i s t i c c ond i t i on .

If f ( u ) has no infle c t i on point s , then impli e s

Suppo se , e . g . , that fuu

0 ; then f ' a is an inc re as ing func t i on

of u , so imp lie s that uz

ur

. S inc e d i ffe renc e quot i ent

al s o are inc re as ing func t i ons of the i r argument s , foll ows .

When f has infle c t i on p oint s , the s t ab i l i ty c ondi t i on is a

genuine ad di t i onal re s t r i c t i on .

It c an be shown that eve ry ini t i al value p rob lem for a s ingle

c ons e rvat i on law has a uni que s olut i on in the int egral

s ens e that exi s t s for all t ime t O and all who s e dis

c ont inui t i e s are s t ab le in the s ens e of In Se c t i on 7 we

shall pr ove the uni quene s s o f such a s olut i on , and c ons t ruc t s olu

ti ons wi th p i e c ewi s e c on s t ant ini t i al dat a .

We turn now fr om s ingle c ons e rvat i on laws t o sy s t em s . He re

N(we have not one but N s ignal s pe ed s u ) . We c laim

that the appropri at e extens i on o f the charac te r i s t i c c ond i t i on

t o thi s c a s e is

The re exi s t s an index k , l k N , s uch that

ak<u,) s ak tur )

whi le

( uz) S 8 .

r

A d i s c on tinui ty s at i s fy ing thi s c ond i t i on i s c alled a k- shock .

The le ft half of the two ine quali t i e s s ay s that exac t ly N- k+ l

1 — 2 0

c harac t e r i s t i c curve s imp ing ing on the d i s c ont inui ty from the

le ft ; the r ight half of the ine quali t ie s s ay that exac t ly k imp inge

f rom the right . So alt oge the r the t ot al numb e r of charac te r i s t i c s

that impinge on the d i s c ont inui ty fr om e i the r s ide is N+ l . E ach

of the s e charac te r i s t i c s c ar r i e s one pi e c e of info rmat i on ; the s e

N+ l data , c omb ine d wi th the N- l re lat i on s that c an be ob tained

f rom the N R- H c ond i t i on s by e lim inat ing s , are nee de d t o

de te rm ine , i t e rat ive ly , the 2 N c omp onent s of u and ur

.

ZWe pr oc e e d now t o show that s olut i on s whos e d i s c ont inui t ie s

vi olat e the charac t e r i s t i c c ond i t i on s c an be s pli t into

rare fac t i on wave s , and thus are in thi s s ens e un s t ab le . The re fore

s uch s olut i on s are re j e c te d a s not re ali z ab le phy s i c ally .

The c ons e rvat i on laws in b oth di ffe ren t i al and inte gral

f orm are invari ant unde r a un i form s t re t ching of b oth the x and t

variab le s ; i t f ol lows that has s o- c al le d c ent e re d s olut i on s ,

i . e . s o lut i ons that de pend on x/t alone . We shall de s c r ib e now

the s e s olut i ons ; the re are two kind s : shocks and rare fac t i on

wave s . A shock i s of the fo rm

ufl

f or x/t s

u ( x , t )

ur

f or s x/t

whe re the s t at e s u

zand u

rs at i s fy the R - H c ond i t i on s We

ask : given uz, de s c r ib e the s e t of s t at e s u

rthat c an be c onne c t ed

t o ufl

through a s ingle shock . Thi s i s e as i ly answe re d i f we are

looking fo r we ak shoc ks , i . e . s t at e s ur

c l o s e t o ug

. We c laim that

they form N one - parame t e r fam i li e s ur

u ( s ) ; we take now the jump

re lat i on

2 0

1 - 2 1

s [ u - u£] f ( u ) - f ( u

2)

and d i ffe rent i ate twi c e , wi th re s pe c t t o e . Denot ing d/d e by

and re c all ing

gra duf A

s'[ u ] -+ su

' Au '

s [ u ] + 2 s ‘u' + su

"Au

"+

O in ( 3 . 2 9 ) lwe ge t

su' Au '

whi ch shows that s ( 0 ) is one of the e igenvalue s

II "S

s:u' ( 0 ) the c or re sp onding e igenve c t or r

and s e t t ing u' r , s a we ge t

2 s 'r i— a u

"Au

"-t A '

r

Now take the e igenvalue re lat i on

a r Ar u u ( e )

and di ffe rent i ate at s O

a 'r -t a r' A r ' +- A '

r

Sub trac t ing thi s from give s

( 2 8 ' - a ' ) r -+ e ( u - r'

) A ( u r

2 1

1 - 2 2

Mul t i ply ing thi s wi th the le ft e igenve c t or z o f A give s

( 3 -3 2 ) 2 s ‘

Now

u’ o

g ra dua r -gradua

Le t 's as sume that r -gradua 0 , then

r -sr a dua

c an be achieve d by no rmal i z ing r . I t

and u' r that

1 a'

Not e that when r i s no rmali z ed by the parame t e r a ne e d s

re s c aling in orde r t o have u' r . I t is easy t o s e e that , fo r 6

small enough , the charac te r i s t i c c ond i t i on is s at i s f ie d i ff

s i s negat ive .

S inc e a i s any one of N e igenvalue s , we s e e that N one

parame t e r fam i li e s ur

c an b e c onnec te d t o uzby a s ingle shock ;

exac t ly one hal f of e ach fam i ly s at i s fi e s c ond i t i on

We turn now t o rare fac t i on wave s ; di ffe rent i ab le s olut i ons

o f of form

a ( x , t ) w ( p ) p x/t

Sub s t i tut ing thi s int o we ge t

- pw' i- Aw ‘ O

whi ch is s olve d by

2 2

https://www.forgottenbooks.com/join

b e c onne c te d t o a one - parame t e r fami ly of s t ate s u2 ( el , 6

2 )thr ough

a wave pe r t aining t o the s e c ond wave s pe e d a2, e t c . C ont inuing in

thi s fashi on we s e e that by go ing through all avai lab le wave s we

c an c onne c t any s t ate uO

t o an N parame t e r fami ly of s t at e s

duuN

uN( e ) . We have shown ear li e r tha t '

HE;rk;

s inc e the right e igenve c t or s rk

are l ine ar ly inde pendent i t fol l ow s

that f or 8 small the fam i ly uN( e ) s imp ly c ove r s a full ne ighb o r

ho od of uo

. Thus we have shown

Supp o s e c ond i t i on hold s ; then given any two s t at e s

u and u s uffi c i ent ly c lo se , the re exi s t s a s olut i on u ( x , t ) ofo n

wi th ini t i al value s

u ( x , o )

Thi s s olut i on i s c ente re d , i . e . a fun c t i on of x/ t , and c on s i s t s of

N+ l c on s t ant s t ate s s e parate d by shocks or c ente re d rare fac t i on

wave S

Figure

An init i al value pr ob lem of form wi th ini t i al dat a c on

s i s t ing of two c ons t ant s t at e s , is c alle d a Riemann ini t i al value

pr ob lem .

C ondi t i on i s a kind of a c onvexi ty c ond i t i on ; the re

are s ome impor t ant c as e s whe re i t fai l s t o hold , e . g . for so

c alle d c ont ac t d i s c ont inui t i e s r -gradua O ; in thi s c as e the c on

c e p t of s hock and rare fac t i on wave c oale s c e and the re s ul t s t i ll

hold s .

For what sy s tem s d oe s the ab ove re s ul t hold in the large ?

that i s , i f we do not re s t r i c t the parame t e r s 6 t o be small , how

large a ne ighb orhood of uO

i s c ove re d by u ( e ) , and i s the c ove ring

s imple ? i f not , the ini t i al value prob lem ha s s eve ral s olu

ti ons and we nee d s ome c ri t e ri on in ad di t i on t o to i dent i fy

the phys i c ally re ali z ab le s o lut i ons .

Be the and Wey l have shown , s e e l ] and that i f p i s a

c onvex func t i on (if p

' 1at c ons tant entr opy , then the init i al value

p rob lem ha s only one s o lut i on . Wendr off ,has

inve s tigate d the s i tuat i on when thi s c onvexi ty c ond i t i on i s

vi olate d .

In an inte re s t ing s e quenc e of pape rs Liu has

analy z e d the Riemann ini t i al value prob lem when is vi olate d ;

he has de rive d an analogue o f c ond i t i on fo r sy s tems , and

has appli e d i t t o the e quat i ons o f gas dynami c s .

inte re s t e d in approximat ing s o lut i ons o f evolut i on

the s c hemat i c fo rm

app roxima te solut i on ope ra t o rs Sh( L ) whi ch , when

25

appli e d t o the ini t i al value uO

u ( O ) of a s olut i on u of

furni she s an app roximat i on t o the value o f the s olut i on u at h

S ( L ) ( u u ( h ) +— e rr orh 0

He re h i s a small quant ity ; t o app roximat e u ( t ) , t not small , the

ope rat or S i s appli e d re peat e dlyh

( 4 . 2 )N

u ( t ) -+ e rr or t Nh

The appr oximat ing ope rat o r S L ) i s c on s t ruc te d s o that the e rr orh(

in i s small ; thi s impli e s that the e r r or in i s al s oN

i f , and only i f , the s cheme i s s t ab le in the s ens e that Sfinot magni fy . If thi s c ondi t i on is fulfille d , then the e rr o r in

( L ) doe s

( 4 . 2 )Nis , r oughly , N t ime s the e rror in Thi s shows that we

mus t choo se S s o that the e rr or in i s O ( h2) .

h

The e rr o r in c an be apprai s e d by Tay lor ‘ s the o rem

u ( h ) u ( o ) + hut ( o ) + O ( h2) u

o+ hL ( u

O

Thi s show s that in orde r t o make the e r r or in O ( h2) , S L )

h(

mus t s at i s fy

Sh( L )

In many p rob lem s , par exce llence in flui d dynam i c s the ope rat or L

is the s um o f seve ral ope rat or s Li’ each de s c r ib ing a d i ffe rent

phy s i c al me chani sm

L =Z Li

2 6

ope rat or for a s the p roduc t of app roximat e s olut i on ope rat or s

S ( L of the part i al e quat i onsh i

ut

Li ( u )

E ach Sh( L

ls ati s fi e s

2( 4 6 )

lSh( L

l) — ZI + rfl

fi4— O ( h

We s e t

sh( L ) T

—

TSh( Li)

It is e asy t o show that i f e ach Lis at i s fi e s then S

h( L )

1

de f ine d by s at i s fie s with L 5 Li

’

The me thod of frac t i onal s te p s has s eve ral d i s t inc t

advantage s

1 ) E ac h e quat i on us ual ly ha s i t s own s pe c ial fe aturei

( symme t ry , invari ance , e t c . ) whi ch can be exploi t e d t o c ons t ruc t

an e ffi c i ent s cheme Sh (Li) .

i i ) If e ach s cheme Sh ( Li )

i s s t ab le in the s en se that i t doe s

not inc re a s e s ome norm , such a s the L2

norm , c ommon to all e qua

t i ons then likewi s e the p roduc t d oe s no t inc re as e

that no rm and s o i s aut omat i c ally s t ab le . E ven i f e ach Sh ( Li )

inc re as e no rm s light ly

ushuin l + o( h )

i t only c aus e s a s im i lar s light norm inc re as e by Sh ( L )

. Thus

ins t e ad of having t o che ck the s t ab i li ty of a compli c ate d compos i te

27

s cheme i t suffi c e s t o che ck the s t ab i li ty of e ach of it s fac t or s .

i i i ) Prog ramming c onveni enc e : one c an wri te a pr ogram for

imp lement ing a s cheme of fo rm whi ch c ons i s t s of n di s t inc t

package s s t rung t oge the r in s e ri e s , e ach package s o lve s an e quat i on

of form If one want s t o inc orporate an impr ove d me thod

for s olving the ith

e quat i on , only one of the package s has t o b e

i.

rewri t t en .

iv ) Ye t anothe r advant age i s de s c r ib e d in Se c t i on 5 .

Re lat i on s ay s that apply ing the app r oximat i on s cheme

onc e le ad s t o an e rr or of s i z e O ( h Re pe at ing the appr oximat ion

N t ime s , whe re Nh T final time re sult s in cumulat ive e r ror of

s i z e N0 ( h2) O ( h ) , pr ovi de d that the s cheme i s s t ab le . To b ring

t he e rr or down t o ac c e p t ab le s i z e may re qui re making h s o small

that the t im e re qui re d t o pe rf o rm N T/h s t ep s i s unac c e pt ab le .

In thi s c as e i t is p o s s ib le t o reduc e the numbe r of s t ep s re qui re d

by emp loy ing a s cheme that is ac curat e t o s e c ond orde r , i . e . that

appr oximate s u ( h ) wi th an e rr or O ( h3) . We exp lain how t o do thi s

in c as e the ope rat or L is line ar ; we allow L t o de pend on t .

We s t ar t wi th Tay lo r 's the o rem t o s e c ond orde r

h2

u ( h ) u ( o) + hut (0 ) T ut t ( o )

+ o( h3)

The fi rs t de rivat ive o f u i s given by the s e c ond c an b e

ob t aine d by di ffe rent i at ing wi th re s pe c t t o t

2ut t

Lutd- L

tu ( L — L

t) u

Sub s t i tut ing int o the Taylo r approximat i on give s

2 8

1 - 29

h2

u ( h ) ‘

2 [ L2( O ) + Lt

Thi s shows that for s e c ond orde r ac c uracy we mus t have

(a 9 ) s ( L ) 1 + m o ) ( o) ] + o( h3 )h 7? t

The re ade r may e as i ly c onvinc e him s e lf , in the s imple c as e when

Li

are inde pendent of t , that even i f e ach Si

i s a s e c ond orde r

appr oximat i on t o

s ( L . ) I + hLi( O ) T

L

the pr oduc t is s t i ll only a fi r s t orde r approximat i on t o

unle s s all the Li

c ommut e ( whi ch they don 't in gene ral ) .

G i lbe rt S t rang , has devi s e d a var i ant of the me thod whi ch

d oe s not suffe r from thi s re s t ri c t ion ; we shall de s c ribe i t for

t e rm s , i . e . when L is of the form

L == A + B

The o rem ( S trang ) : Suppo s e Sh( A ) and S

h ( B ) are s e c ond orde r

approximat i ons t o

u Au and u Bu

re s pe c t ive ly . Then

Shh. )

i s a s e c ond orde r approximat i on t o s olut i ons of

The proof i s a s imple mat te r of alge b ra : B y have ,

modulo te rms O ( h3)

2 9

h h 2B ( O ) ‘

s [ B + Bt]

h2

2I + hA ( O ) 7 [A + A

t]

S I +hE (h)

h?

[ 82+ B ]

h/2 2 2 2‘

s‘

t

h h2

2B ( O ) “

a” [ B + 3 B

t]

The t ri ple pr oduc t i s , mod O ( hz) ,

h2

2 2 2 132

g. [ A + A

t+ZB + 13

t 2BA + AB ]

and thi s by i s indee d e qual t o

h2

2I + hL ( O ) g [ L + L

t]

We turn now t o nonline ar e quat i ons of the form whe re

the flux f is the sum of fluxe s fJ

ut L

j

d iv fJ. s

The analogue of the me thod o f frac t i onal s t e p s c on s t ruc t s a ppr oxi

mat i ons t o s o lut i ons of as a p roduc t of ope rat ors a pproxi

mat ing s olut i on s of the part i al e quat i on s

(u, l4 )i

ut

-t div fj

O

and

( 4 -1 5 ) ut

S

In prac t i c e the e quat i ons ( 4 . l4 ) iare oft en furthe r de c ompo s e d as

3 0

https://www.forgottenbooks.com/join

ave rage flux f be twe en t ime s t e p n and n+ 1 /2 at the b ound ary b e

twe en Ik

and I The appr oximat i on t o the n i sk+ 1

‘

n+ 1 n T n+ 1 /2 n+ 1 /2Vk

- Vk g g

k- l/Q) 0

whe re T is the t ime s t e p fr om n t o n+ 1 . S inc e the flux f i s a

n+ 1 2func t i on of the dens i ty u , we t ake the app roximat e f lux gk+ l§2 t o

b e a func t i on of the approximat e dens i t i e s at a fini t e numb e r of

po int s ne ar the p oint k+ l/2 and t ime n+ 1 /2

n+ 1 /2 n n n+ 1 n+ 1gh+ i/2

g<Vk

We re qui re g t o be c ons i s tent wi th f , in the s ens e that

( 5 5 ) f ( u )

It i s c onvenient t o re gard v a s b e ing de fine d for all x,

t

nV ( x , t ) V

kfor x in I

ktn

t tn+ b

We show now that the c ons i s t ency c ond i t i on guarant e e s that

i f a s e quenc e of s olut i ons of wi th ini t i al value s uo( x )

tend s a s 6 , t- O b ounde dly and almo s t eve rywhe re t o s ome func t i on

then thi s lim i t u i s a s olut i on o f the integ ral form

of the c ons e rvat i on law wi th ini t i al value uo( x ) . Fo r le t w ( x , t )

b e any smo oth t e s t fun c t i on whi ch is z e ro for x , t l arge ; mult ip ly'

by sum ove r k and n and sum by part s ; we ge t

n n— w

Z kVk k k+ l m+ l/2

3 2

+ 1 2I f vit end s t o u b ounde d ly and a . e .

, then gi+ l§2 , de fine d by

and s at i s fying tends t o f ( u ) , b ounde dly and a . e .,

and

tend s t o

(w v -

tf ) dxdt O

t

Thi s i s pre c i s e ly re lat i on

He re are s ome examp le s of appr oximat i ons t o flux fun c t i ons

n+ 1 /21 ) g

k+ l/2—

e

n+ 1 /21 1 ) gk+ l/2

f

m+ l/2 f ( uk

-t f ( uk )i i i ) gh+ i/2

nf (

n) + f (

The following ob s e rvat i ons are obvi ous but us e ful .

a ) If g is c ons i s tent wi th f , and i f h is a func t i on o f vn

that vani she s when all it s argument s are e qual , then g+ h t oo i s

c ons i s tent wi th f . For example we c an augment the app roximat e flux

fun c t i on in i ) t o

f (n) + f (

V ) 82112 M i

anothe r c on s i s t ent flux appr oximat i on .

b ) Supp os e the flux f i s the sum of s eve ral fluxe s

fli- f

2+ f , and s uppo s e we tre at the s e fluxe s by the me thod of

frac t i onal s te p s explained in Se c t ion 4 . I f a t each s tep we employ

a f lux appr oximat i on c ons i s t ent w ith the part i al flux fi, then the

ove rall s cheme wi ll b e c ons i s t ent wi th the t ot al f lux f . Thi s

app li e s in par t i c ular t o the import ant c a s e when the numb e r of

s pac e var iab le s i s gre at e r than 1 , and the flux ha s an x and a y

c omponent .

c ) The flux app roximat i on ha s t o be s o cho s en that the s cheme

is s t ab le . Our examp le s i ) and i i ) are uns t ab le , i i i ) is

s t ab le , iv ) i s s t ab le i f exc e e d s the s ignal s pe e d s , and v ) is

s tab le i f c i s large enough and i s small enough .

6 . The method s of G odunov and Glimm

The s e me thod s we re devi s e d for c ons e rvat i on laws in one s pac e

variab le

ut

-l—fX

0

As in Se c t i on 5 we d ivi de the x- axi s int o c e ll s I e ach of lengthK

)

6 ,c ente re d at x k6 , se e Given any ini t i al dat a u

o( x ) we

c an pr o j e c t i t ont o the s pac e of func t i on s whi ch are c ons t ant on

e ach I by s e t t ingk

vo

( x ) uo( x ) dx x in I

k

We de fine the func t i ons a s the s olut i on of e quat i on

wi th the fol lowing ini t i al value s

v for x

for x

1 -3 5

Thi s is a Riemann ini t i al value prob lem of form it s s olu

t i on c ons i s t s of N+ l s t at e s s eparate d by N wave s c ente re d at

x t 0 . E ac h wave t rave ls wi th a spe e d that e qual s

or i s b ounde d by one of the s ignal s pe e ds a . Deno te the maximum

max ;i t fol lows then that the c ente re d wave s

i s s uing from two ad j ac ent cente r s ( k- l/2 ) 6 and don 't

s ignal spee d by a ]

inte r s e c t e ach othe r as long a s

6

So during the t ime inte rval the s olut i ons Vk+ l/2

wi th

ini t i al value s c an be fi t te d t oge the r t o form a s ingl e exac t

s oluti on V ( x , t ) of wi th ini t ial value v0

given by

Thi s s olut i on c on s i s t s of c on s t ant s t at e s s eparat ed by c ente re d

wave s , s ee Fig .

F igure

Af te r t ime the wave s i s suing fr om ad j ac ent cente rs s t art t o

in te ra c t; in a nume ri c al me thod deve loped in the fi ft ie s

Godunov re plac e s v ( k, r ) at r 6/2 | a | maxby i t s pie cewi se c ons tant

pro j e c ti on de fined by No te tha t the inte grat i on ind i c ated

by nee d not be c arrie d out expl i c i t ly ; fo r V ( x , t ) i s an exa c t

s olut i on of so the integ ral fo rm of give s

3 5

V ( X , T ) dx v ( x , O ) dx f ( v ( ( k

Thi s c an be rewri t ten as

Vi Vi giZi/2 )whe re

eifi/g

s in c e b e ing c ente re d at i s independent of t .

1Onc e v has b e en de t e rm ine d as a pie c ewi s e c ons t ant func t i on ,

the b as i c s t ep i s re pe ate d ; thi s i s done as many t ime s as ne c e s s ary

t o re ach the t ime s T at whi ch the phenomena unde r inve s t igat i on

are t aking p lac e . We denote by the int e rme d i ate t ime s

at whi ch the pro j e c t i on s take plac e .

o o 0,Note

zhzt if v

kVk+ l

’ Vk

’

have then gk+ l/2

f ( vfi ) ; thi s p rove s that the appr oximat e flux

then v by we

emp loye d in Godunov 's s cheme i s c on s i s t ent wi th the exac t

f lux f .

Glimm 's me thod re s emb le s G odunov ‘

s ina smuch a s the a pproxi

mat i ons V ( x , t ) employe d are p ie c ewi s e c ons t ant func t i ons of x at

the s e le c t e d t ime s and are exac t s o lut i ons in the s t rip s

tn— l

t tn’ and are di s c ont inuous ac r o s s t t

nHoweve r G limm

de fine d v di ffe rent ly at t ime tn; ins t e ad o f G l imm s e t s

V ( k6 + an6 , t

n

- O ) x e 1

whe re denote s the limi t ing value s of v a s t t fr om

vari ab le s chos en fr om a s amp le uni fo rm ly d i s t r ibut e d in

Glimm shows that for almo s t all choi c e s of [ on

] ,

v c onve rge s t o an exac t s olut i on a s 6 ' 0 . He re is a

s l ight ly mod i fie d form of his argument .

Le t w ( x , t ) be a sm ooth te s t func t i on , o for | x | large .

Mul t ip ly vt

-r f ( v )X

0 by w , and integrate by part s in the s t rip

tn- l :.

t f-tn; we ge t

— O ) - w ( x , t )v ( x , t + O ) ] dx

[ wtv O

tn- l

ove r 0 n M; denot ing tM

T we ge t

[ wtv

r

whe re

( 6 9 ) rn

wi th

- O )

Lemma Deno te n an uppe r b ound fo r the t ot al vari at ion

V ( x , t ) as func t ion x . Then

rn

c ons t 6n| w |max

Proof : Denote the vari at i on of ove r the uni on

and IkJr1

by nk+ l/2' Sinc e v i s c on s t ant ove r e ach I

k’

Z T1h+ i/2 2Ti

Glimm s hows that for x in Ik

| v ( x , tn- O ) c on s t (”k

for de t ai le d p roof we re fe r t o [ 9 he re we me re ly ob s e rve that

i f nk+ l/2and qk- l/2

are b oth z e r o , then v ( x , t + 0 ) has the s amen - l

Ivalue in al l thre e int e rval s I I s o that v ( x , t ) i sk— l

’ k ’k+ 1

’

c ons t ant in Ik

for tn- l

t tn+ l

’ and the re f ore v ( x , tn

- O )

in 1k

“ Thus i f the right s ide of is z e ro , s o is

the l e ft s i de ; whi le thi s doe s not pr ove ine quali ty i t

m ake s i t plaus ib l e .

We mul t ip ly by w ( x , tn) and int egrate ove r I s inc e

k ;

the length of I i s 6 we ob t aink

| v ( x , tn- O ) dx

c ons t 6 ( T1k- l/2w

max

C omparing and we c onc lude that

lrn| c ons t 6 | w max

2 z nk+ l/22 c ons t 6n| w max

whe re in the las t s te p we us e d Thi s pr ove s

3 8

https://www.forgottenbooks.com/join

- O )

2c on s t ( l - l/2 Iw

xlma x

Summing ove r all k we de duc e fr om us ing that

holds .

Lemma Fo r m n ,

rmrnda c ons t 6

3n2| w | W

max x max

whe re do

Proof : Supp os e m n ; then rm

i s independent of on; s o we ge t ,

us ing and that

nr r du

nr dd 6

2q | wc ons t 6n| wm n n max x max

Integrat ing wi th re s pe c t t o the re s t of the dJ yie ld s ine quali ty

We are now re ady for the main e s t imate ; us ing and

we ge t

M

rgda ( s;

rn)2dct z: r r dd

M

E riddl n

c ons t M62q2| w ia x + c on s t M

203n2| W |max W

X max

M i s the numbe r of t ime s t ep s , 6 the s i z e of the s pac e s tep . S inc e

the s i ze of the t ime s tep , sub j e c t t o ine qual ity i s taken

4 0

t o be a s large as po s s ib le , i . e . a lmax ’ we have

T (3M E

l

tn

- tn_l 3

M6 /2 Ia lmax

Se t t ing thi s int o we ob tain the e s t imat e

rgda co

whe re

2C c ons t T] | w max'a‘ max

wmax

+ T | a max X max

In [ 59 ] Glimm e s t imat e s for all t the t ot al vari at i on of the

appr oximate s olut i ons v for all c ho i c e s of a in te rm s of u t ot al

vari at i on of the ini t i al dat a . Thi s give s an e s t imat e for n vali d

for all a .

The quant i ty r , de fine d by i s the amount by whi ch the

appr oximat i on v fai l s t o s at i s fy the int e gral form of the

c ons e rvat i on law . We c all r the re s idual , wi th re s pe c t t o the te s t

func t i on w .

Given any 5 , i t follows fr om that

| r | < e

exc e pt po s s ib ly fo r an a - s e t of meas ure C6 c- 2

. Given N c ons e rva

t i on laws and K te s t func t i ons all re s iduals are 6

exc e p t pos s ib ly on a s e t whi ch i s the un i on of the exc ept i onal

s e t s fo r e ac h ind ivi dual w and e ach c ons e rvati on , and whi ch may

the re fore have as large a me asure as CKN6 e- 2

; thi s e s t imate i s

unduly pe s s imi s t i c , allowing for no ove rlap among the exc e pt ional

s e t s .

What are the imp li c at i on s for re al i s t i c value s of the

parame te r s ? Le t 's t ake the c as e that the tot al variat i on of the

ini t i al dat a i s 1 ; then we c annot expe c t a b e t te r e s t imat e f or n

than n 1 . Le t 's t ake a t e s t func t i on w Wi th [w ]max x max

Suppo s e the maximum s ound s pee d | a | maxl , and le t 's t ake a s

final t ime T l . A re al i s t i c value for the c ons t ant in ine qual i ty

i s 1 . Se t t ing all the s e numb e r s int o give s C 2 ,

s o we c onc lude , for a s ingle c ons e rvat i on law and te s t func t i on ,

that

| r 6

exc ep t po s s ib ly on a s e t of a o f me as ure 2 6 2- 2

. It is not uh

re as onab le t o want t o make 2 1 0- 2

,and ord inary prudenc e re qui re s

making the me as ure of the exc e p t i onal s e t le s s than 1 0— 2

. To

s at i s fy thi s we mus t have

2 6 1 02

i . e . the s pat i al s t e p s i z e 6 mus t b e le s s than 5 >< lo- 5! Thi s is

ve ry fine g ri d , hard ly c alle d fo r t o ac hieve a re s olut i on whi ch ,

wi th | w | [W I 1 , i s of the orde r o f uni ty .

max x max

It i s i lluminat ing t o exam ine how Glimm 's s cheme tre at s a

part i cular ly s imp le Riemann ini t i al value p r ob lem

u ( x , o )

u for O x

whe re uz

and ur

are s o cho s en that the exac t s olut i on c on s i s t s of

a s ingle shock wave p ropagat ing w i th s pe e d s

4 2

u ( x , t )

For thi s c al culat i on i t i s c onveni ent t o t ake Ik

wi th thi s choi c e Gl imm 's s cheme re ad s

v ( k6 +— 8n6 , t

n

- O )

whe re an

and- l/2 , n Note that the B '

s are uni formly

di s t ribute d in

Sinc e v ( x , t - 0 ) i s the exac t s olut i on we ge t f rom

that

u

zfor x J

l6

v ( x ,+ 0 )

u for J16 x

whe re

Re peat ing thi s M time s we ge t

V ( X, MT )

of SJ ST/5

the app roxim at e t ime

6

N TT' “

ifT

As M mi , JM/M de fine d by tend s t o sr/b , s o the shock

loc at i on t end s t o E -T- T sT, the exac t loc at i on of the shock at

6 Tt ime T . A s imp le c alculat i on shows that the expe c t e d devi at i on

o f JM/M f rom i t s expe c te d value K i s cA/M ,

whe re

c So us ing and T Mt we s ee that the

expe c t e d devi at i on of the c alc ulat e d value of the s hock po s i t i on

fr om the t rue one i s

O ( T/T ) 6

Le t ’s t ake T 1 , 6 /T 1 , s the n 2 To make the

expe c t e d dev i at i on s , we mus t have

6 87re2

For 2 1 0- 2

th i s me ans 6 2 o 5 >< lo- 3

. Thi s i s not t o o b ad but

ge t s wo r s e a s T inc re as e s .

Note that the ac c uracy of G limm 's s cheme app li e d t o the

s pe c ial Riemann pr ob lem ab ove c an be inc re as e d app re c i ab ly by

t aking the s e quenc e Bn

not at rand om but a s uni form ly di s t ribute d

as po s s ib le . F rom the po int o f vi ew o f e qui d i s t r ibut i on an a t tr a c

t ive choi c e i s an

ne (mod whe re 8 i s an algeb rai c numbe r ,

s ay / 2fl The e rr or in shock po s i t i on when app li e d t o the s pe c i al

l og N)Riemann p rob lem ab ove i s O ( N

The us e of such s e quenc e s in

Monte C arlo c alc ulat i ons ha s b een s ugge s t e d by R . D . Richtmye r in

the e arly 5o's , and in c onne c t i on wi th Gl imm'

s s c heme by the

author , Chorin has suc c e s s ful ly int r oduc e d o the r type s of

we ll d i s t r ib ute d s e quenc e s .

Re cent ly Tai Ping Liu suc c e e de d in showing the c onve rgence

of Glimm's s cheme t o a s olut i on for equidis t ribut ed s e quenc e s when

the ini t i al d at a are arb i trary . The t ime rate of c onve rgenc e is

an open pr ob lem ; i t s de te rminat i on wi ll have to b e , mo s t like ly,a

c omb inat i on of the ory and nume ri c al expe rimentati on .

Godunov has suc c e s s ful ly app li e d hi s me thod t o sy st em s of

c ons e rvat i on laws in s eve ral s pac e vari ab le s by us ing the me thod

of frac t i onal s te p s . Glimm's me thod has b een app li e d by Chor in

in s eve ral spac e vari ab le s , again us ing the me thod of frac t i onal

s t e p s . No analyt i c al re s ult s are avai lab le in thi s c as e .

A . Har ten has ob s e rve d that when Glimm ‘s me thod is us ed in frac

tiona l s t e p s t o c alculate the pr opagat i on of a c ont ac t disc ontin

uity in two d imens i ons , the re s ult ing one d imens i onal pr ob lem s are

re s olve d in t e rm s of shock s . Thi s int roduc e s a c e rt ain amount of

exc e s s ent ropy produc t i on .

In Se c t i on 3 we s aw that s eve ral s olut i ons in the integral

s ens e of a sy s tem of nonline ar c ons e rvat i on laws c ould have the

s ame init i al value s . S inc e the ini t i al c onf igurat i on ought t o

de te rmine the flow in the future , only one of the s e s eve ral s olu

ti ons c an oc cur in nature , and all othe r s have t o be exc luded on

the bas i s o f s ome phys i c al o r mathemat i c al princ i ple . In Se c t i on

3 we have fo rmulate d two s uc h princ i ple s

i ) s tab i li ty , i i ) the charac te r i s t i c c ond i t i on .

In thi s s e c t i on we fo rmulat e two furthe r princ i ple s , and show

tha t , in s uffi c i ent ly s imple c as e s,all four are e quivalent .

45

We s tart wi th the fo l lowing que s t i on : i f u i s a sm ooth s olu

t i on of the sy s tem of c on s e rvat i on laws

ut+ f ( u )

Xo

d oe s u sat i s fy s ome othe r c ons e rvat i on law that i s not me re ly

l inear c omb inat i on of the e quat i ons ( 7 To answe r thi s we

in the d i ffe rent i al fo rm

ut

O A grad f

Le t U U ( u ) be s ome fun c t i on of u ; mul t i p lying grad U

( U N )we ge t

u 11

Ut+ grad U A u

X0

I f the re i s a func t i on E ( u ) such that

grad UA grad F

then c an be wri t ten as a c ons e rvat i on l aw

Ut

-rrx

0

S inc e in our de rivat i on we us e d the d i ffe rent i al form o f the

e quat i on , we c annot c onc lude that a s o lut i on of in the

int egral s ens e s at i s fi e s in the integral s ens e ; in fac t , as

we shall s e e , the oppo s i te of thi s i s t rue .

We remark that i s a sy s tem of N line ar di ffe rent i al

e quat i ons for the two func t i on s U and F . For N 2 the re are

p lenty of s olut i ons ; for N 2 the re are none in gene ral , exc e pt in

s pe c i al c as e s . For examp le,Godunov has ob s e rve d that when A i s

46

https://www.forgottenbooks.com/join

Le t 's rewri te in nonc ons e rvat i on form

Suppo s e U i s an ent r opy func t i on ; then mul t iply ing by

grad U we ge t , us ing

Uti- F

Xx grad U u

Us ing the chain rule we ge t , d i ffe rent i at ing UX

grad U ux, that

TUxx

g rad U uxx

+ uq uux

Sinc e U i s an ent r opy func t i on , i t i s c onvex , i . e . the mat r ix of

it s s e c ond de rivat ive s is p o s i t ive de fini te

Uuu

0

We de duce from and that

g rad U uxx j -

Uxx

Sub s t i tut ing thi s int o we ge t

Ut+ F

XAU

XX

Suppo s e that u ( A ) i s a s e quenc e of s olut i ons of that te nd s

as A 0 b ounde d ly and a . e . t o a l im i t u . Then U ( i ) and

F ( A ) tend t o U and F in the s ens e of d i s t ribut i ons ,

whi le the right s i de of tend s t o z e r o in the s ens e of

di s t r ibut i ons . So we have p r ove d the

4 8

0

Suppo s e u is a pie c ewi s e smooth s olut i on wi th d i s c ont inu

i t i e s ; then U i- Fx

O in the sm ooth re gi ons , whi le on a di s c ont

tinuity x x ( t )

Ut+ F

x6 ( x - U

r] - [ F

zFI"

We draw two c onc lus i ons fr om thi s

O

Deno te by U ( t ) the t o t al ent ropy at t ime t

( 7 -1 6 ) U ( t ) U ( x , t ) dx

then

“t Z s [ U£

- Uri [ F

zFr]

Thi s shows that the l e ft s i de of is the rat e at whi ch

ent ropy is d im ini she d at the d i s c ont inui ty . and

The vi s c os i ty the o rem charac t e ri z e s s oluti ons of that

are lim i t s o f vi s c ous s olut i on s wi thout c arry ing out the l im i t ing

pr oc e du re ; the c harac te r i z at i on i s in te rm s of ent ropy . We c onne c t

now thi s ent ropy c ondi t i on to the s t ab i li ty c ond i t i on s t ated in

Se c tion 3

1 - 5 0

are s tab le in the s ens e of c ond i t i on

Proof , due t o Hopf [ 1 6 ] and Kru z kov , [ 1 9 ]

Le t be a d i s c on t inui ty of u , s ay uz

ur; le t v be

any value b e twe en the two

We de fine U by

U ( u )

No te that U i s a c onvex func t i on . Se t U int o the di ffe rent i al

e quat i on ( 7 we ge t

Inte grat ing give s

Wi th thi s c hoi c e of U and F we have

Ug

" Ur

v - ur

Fg

‘ Fr

fv — fr

Se t thi s,

t oge the r wi th de fini t i on of s , int o afte r

a li t t le re ar rangemen t we ge t

5 0

LI - V v - u

f ( v )u

r

_uf, u A } f

r1"

l"

whi c h i s e quivalent wi th c ond i t i on when uz

ur

. The c as e

u ur

c an be re duc e d t o the previ ous c as e by us ing the c ons e rva3

t i on law , ent ropy and s t ab i l i ty c ondi t i on s at i s fi e d by - u .

If we c omb ine the vi s c o s i ty and ent ropy the orem s, we de duc e

that the fol lowing s t atement s ab out d i s c ont inuous s olut i on s of

s ingle c ons e rvat i on laws are e quivalent

( 1 ) u is the lim i t of s olut i on s of the vi s c ous e quat i on

( I I ) u s at i s fi e s the ent r opy c ond i t i on for eve ry

ent ropy func t i on .

( II I ) The di s c ont inuit i e s of u are s t ab le in the s ens e of

A d i re c t de r ivat i on of ( II I ) from ( I ) i s c ontaine d in [ 1 8 ]

As remarke d in Se c t i on 3 for c onvex f ( I I I ) is e quivalent

wi th

( IV ) The d i s c ont inui t i e s of u s at i s fy the c harac t e r i s t i c

c ond i t i on

Next we show that d i s c ont inuous s olut i ons that s at i s fy the

s t ab i li ty c ond i t i on are uni que ly de te rmined by the i r ini t i al dat a .

The p roof i s based on the

C ont rac t i on The orem ( Key fi t z , Le t u and v b o th be s olut i ons

o f

ut+ f

XO

and suppo s e that bo th s at i s fy the s tab i li ty c ond i tion then

5 1

u ( t ) - v ( t ) l

Proof : We wr i te

Iu- V I an

( u—v ) dx

whe re the inte rval s In

are cho s en s o that ( u - v ) i s of s ign an

ove r

In

' Denote the end lnt s o f In

a s of c our s e bn

an+ 1

Sinc e u and v de pend on t , s o d o an( t ) and b

n( t ) ; we as sume the

de pendenc e i s d i f fe rent i ab le . Di ffe rent i ate

Iu- v l ZZZen

( ut

- vt) dx ( u - v )

by- f ( U ) V

t x tby and c ar ry ing out the int egraRep lac ing u

t i on we ge t

'

dt |u—v EZZZE n ( v ) - f ( u ) -t ( u -v ) s

whe re s abb revi ate s dx/d t , x an

or bn

’

If an

o r bn

i s a p oint o f c ont inui ty for b oth u and v , then

u v the re and s o the c ont ribut i on t o the r ight s ide of is

z e ro . Suppo s e on the c ont rary that , s ay , bn

i s a di s c ont inui ty o f

u b ut not for v,wi th s ay

( 7 -2 1 ) u ( b

In thi s c as e u - v O in I s o a - 1 . Us ing the de fini t i on

of s and the abb revi at ions

( p 2 2 ) - f + ( u - V )t r

e z uz

- ur

Ac c ord ing t o c ondi t i on

fi

r- f f f

uz

- v - v - u

Thi s and re ad i ly imply that is nonpo s i t ive .

C ondi t i on i s invari ant when u i s re plac e d by - u , f ( u )

by - f ( - u ) , v by - v , f by - f ; thi s prove s that when the ine quali ty

in i s reve rs e d , the c ont ribut i on t o the right s ide of

is s t i ll nonpo s i t ive . S im i larly , c ond i t i on is

invari ant when x is re p lac e d by - x and f by - f; thi s prove s that

the c ont ribut i ons t o the ri ght s i de o f at the lowe r end

point s are likewi s e nonpo s i t ive . Finally , s inc e u and v ente r the

ine qual i ty symme t ri c al ly , c ont ribut i ons t o the ri ght s i de o f

at di s c ont inui t i e s of v are likewi s e nonp os i t ive , as long

as the s e are di s t inc t from the di s c ont inui t i e s of u . If the di s

c ont inui t ie s of u and v inte rs e c t only at di s c re te t ime s , we c on

c lude from that Iu-v l i s a noninc re as ing fun c t i on o f t ; the

exc e pt i onal c as e c an be reduc e d t o thi s by chang ing s l ightly the

ini t i al data o f one o f the func t ions . Thi s c omple t e s the proof of

the c ont rac t i on the o rem .

53

It foll ows that i f u v at t ime t 0 , then u v for all

t ; thi s p rove s the

We turn now t o the que s t i on o f exi s t enc e of s t ab le s olut i ons

w ith pre s c ribe d ini t i al value s ; we c ontent our s e lve s wi th Riemann

ini t i al value s , c on s i s t ing of two c ons t ant s t at e s

w for x O

u ( x , o )

z f or O x

We have remarked e ar li e r that f or ufi

ur

i s e quivalent

wi th the ge ome t ri c inte rpre t at i on of thi s i s that f li e s

ab ove the s e c ant in the int e rval When ug

ur

, c ondi t i on

demands that f li e be low the s e c ant in

To s olve the ini t i al value p rob lem in the c as e , s ay , w z

we c ons t ruc t the c onvex enve lope g of f be twe en 2 and w de fined a s

the large s t c onvex func t i on g whi ch is f , s e e Fig . whe re g

appear s as a d ot t e d line . Deno te by w z u ui

2

Figure

endpoint s of int e rval s whe re g f . Then the s olut i on of the

54

https://www.forgottenbooks.com/join

c ond i t i on is s at i s fi e d .

s at i s fy the R — H c ondi t i ons f orm N one parame t e r fam i l i e s ur( e ) ; the

s t at e s that s at i s fy the s t ab i l i ty c ond i t i on make up half of

thi s fam i ly , c or re s ponding t o e 0 unde r the no rmal i z at i on s

and u' ( O ) r .

Abb revi ate the le ft s i de of by R

R s [ U£

- Ur] - [ F

£- r

r

Obvi ous ly R ( O ) 0 ; we show now that al s o R ' ( O ) 0 . Fo r d i ffe r

ent ia t ing we ge t

R ' ( O )

Mul t i p ly by us ing the fac t that u' ( 0 ) r and the re

fore Au ' ( o ) ar we ge t

a U ' ( O ) F ' ( O )

a ( o) we de duc e fr om the las t two re lat i ons that

o ,

A s t raight forward but s light ly t e d i ou s c alculat i on shows that

O and

l TR ( 0 ) g

r Uuur

S ince the ent r opy U is as sumed t o b e a c onvex func t i on o f u , i t

f ollows that O . Thi s shows that fo r 2 small enough ,

R ( s ) O i ff e O . Thi s pr ove s that for smal l d i s c ont inu i t i e s

s t ab i li ty and ent r opy c ondi t ion s are e quivalent , as as s e r t e d .

56

Not i c e that i f R"

( O ) we re 2 0 then R ( s ) ha s the s ame s ign

for all 2 smal l , regardle s s of the s ign of 2 . The re f ore i f we

as sume the t ruth of the ent r opy the orem , i t follow s w i thout any

t e d i ous c alculat i on that R , the rate of ent r opy produc t i on , is at

m os t cub i c in the shock s t re ngth a .

The c ont rac t i on the orem is mos t like ly not vali d for sy s t em s .

A uni quene s s the orem for a s pe c i al c las s of sys tem s of 2 c on s e rva

t i on laws has b e en given by Oleinik in a more gene ral uni que

ne s s the orem , us ing ent ropy , has b e en given by Dipe rna,

We c lo s e thi s s e c t i on by remarking that the c i rc le of i de as

de s c r ibe d in thi s s e c t i on remains an ac t ive area of re s e ar ch . The

c onc e p t s of ent ropy and s tab i l i ty are c ent ral in de s c rib ing s o lu

t i ons that are limi t s of s olut i ons of e quat i on s wi th vi s c o s i ty ,

re al or art i fi c i al . The charac te r i s t i c c ond i t i on , always ne c e s s ary ,

is no t alway s suffi c i ent and has t o b e supplemente d by the fi rs t

two .

Be the , H ., The the o ry of shock wave s fo r an arb i t rary e qua

t i on of s t at e , OSRD , Div . B , Rep or t No . 545 , 1 94 2 .

Bori s , J . P . and Bo ok , D . L ., Flux- C orre c t e d Trans port . I .

SHASTA , A Flui d Transp ort Algor i thm that Works " , J . C omp .

Phy s ., 1 1 , 1 973 , 3 8 - 69 .

Chor in , A . J . ,

" Random Cho i c e S olut i on of Hype rb o li c Sy s t em sJ . C omp . Phy s . , 2 2 , 1 976 , 5 1 7 - 533 .

C ourant , R . and Frie dr i chs , K . O ., Supe r s oni c Flow and Shock

Wave s , 1 948 , Wi l ey- In te r s c i enc e , New York , re p rint e d bySpr inge r Ve r lag .

Da fe rmos , C .M . ,

" S t ruc ture of s olut i on s of the Riemannp rob lem fo r hyp e rb ol i c sy s t ems of c on s e rvat i on law s " , Ar ch .

Rat . Me ch . Anal . , 53 , No . 3 , 1 974 , 2 03 - 2 1 7 .

Dipe rna , R . J . ,

"E xi s t enc e in the large for quas i l ine ar hype r

b ol ic c ons e rvat i on law s " , Ar c h . Rat . Me c h . Anal . , 5 2 , 1 973 ,

244 - 2 57 .

Dipe rna , R . J . ,

" Uni quene s s of s olut i on s of quas i l inear hype rb ol ic c on s e rvat i on laws " , t o appe ar .

Douglis , A .,

" The c ont inuous de pendenc e of gene ral i z e d s olut i ons o f nonl ine ar part i al d i ffe rent i al e quat i ons uponini t i al dat a" , C omm . Pure Appl . Math .

, 1 4 , 1 96 1 , 2 67 - 2 84 .

Glimm, J . ,

" S olut i on s in the large for nonl inear hype rb oli csy s tem s of e quat i ons " , C omm . Pure App l . Math .

, 1 8 , 1 965 ,

697 - 7 1 5

G l imm, J . and Lax , P . D . ,

" De c ay of s olut i ons of sy s tem s ofnonl ine ar hype rb ol i c c on s e rvat i on laws " , Mem . Ame r . Math .

So c ., 1 0 1

G odunov , S . K . and B ag rynovskii, Y .,

" Di ffe renc e Sc heme s forMany Dimens i onal Prob lem s , D . A . N . 1 1 5 , 1 957 ,

43 1 .

Godunov , S . K .,

On the uni quene s s of s olut i ons of the e quat i ons of hydr odynam i c s Mat . Sb . 4 0 , 1 956 , 467 —47 8 .

Gre enb e rg , J . ,

"E s t imat e s for fully deve l ope d shock s olu

t i on s Ind iana Univ . Math . J . , 2 2 , 1 973 , 9 8 9 — 1 0 03 .

Har low, F . ,

" The partic le — in - c e ll me thod for flui d dynami c sMe thod s of C omp . Phys .

, Vol . 3 ,B . Alde r , e d Ac ad . Pre s s ,

New Y ork , 1 964 , 3 1 9 - 3 43 .

5 8

Har t en , A" The Ar t i f i c i al C ompre s s i on Me thod for C omput ing

Shocks and C ontac t Di s c on t inui t i e s " , C omm . Pure Appl . Mat h .,

xxx, 1 977 , 6 1 1 - 63 8 .

Hopf, E . , The part i al d i ffe rent i al e q uat i on u

t+ uu

xuu

XXC omm . Pure App l . Math .

, 3 , 1 95 0 ,2 0 1 - 23 0 .

Hopf, E .

,

" On the right we ak s o lut i on of the C auchy pr ob lemfor a ua sil inea r e quat i on of fi r s t or de r J . Math . Me ch . ,

1 9 , 1 9 9 , 4 83 -4 8 7 .

Keyfi t z , B ., ( Quinn ) ,

" S olut i on s wi th shocks ; an example ofan L

1c ont rac t ive s emigr oup C omm . Pure App l . Math .

, 24 ,

1 97 1 , 1 2 5 - 1 3 2 .

Kru shkov , N .,

" Re s ult s on the charac te r of c on t inui ty ofs olut i on s of parab oli c e quat i ons and s ome of the i r appli c at i on s " , Math . Zametky , 6 , 1 969 , 97 - 1 0 8 .

Lax , P . D . ,

" We ak s olut i on s of nonline ar hype rb o li c e quat i onsand the i r nume r i c al c omput ati on"

, C omm . Pure Appl . Math ., 7 ,

1 9 54 , 1 59 - 1 93 .

" Sho ck wave s and ent r opy Proc . Symp . Univ .

THE ETT‘

I97 1 , E . H . Za rant one l l , ed ., 6 03 - 634 .

" Hype rb ol i c sy s tem s of c ons e rvat i on laws and themathematic al the ory of shock wave s " , 1 97 2 , S IAM , Phi l .

, Pa .

Lax , P . D . and Wend roff , C ., Di ffe renc e s cheme s for hype r

b o lio e quat i on s wi th hi

gh orde r of ac c uracy"

, C omm . Pure Appl .

Math ., 1 7 , 1 964 , 3 8 1 - 3 9

Liu , Tai - Ping ,

" The ent ropy c ondi t i on and the admi s s ib i li tyof shock Math . Anal . and Appl .

, 53 , 1 976 , 78 - 8 8 .

" S oluti on s in the large for the e quat i ons ofnonisentropic gas dynami c s " , Indi ana Univ . Math . J .

, 2 6 ,

1 977 , 1 47 - 1 77

Mac c o rmack , R . W .,

" Nume ri c al s olut i on o f the inte rac t i on o fa shock wave wi th a lam inar b oundary laye r" , Le c ture No te son Phy s i c s , No . 8 , Spr inge r Ve rlag , B e rlin , 1 97 1 .

Ni shi da , T . and Smol le r , J . A . ,

" S olut i on s in the large fors ome nonl inea r hype rb oli c c on s e rvat i on laws " , C omm . PureAppl . Math . , 2 6 1 973 , 1 83 - 2 0 0 .

Ole inik , 0 . A . , On the uniquene s s of the gene ral i zed solut ionof C auchy 's prob lem for a nonl ine ar sy s tem of equa t ionsoc curring in me chani c s " , Us pehi Mat . Nauk , 73 , 1 957 , 1 69 - 1 76 .

Pe ac eman , D . W . and Ra chford , H . H . Jr .,

" The Nume ri c al So luti on of Parab ol i c and E ll ip ti c E quat i on s , Journ . Soc . Ind .

Appl . Math . , II I , 1 955 , 2 8 - 4 2 .

Rich tmye r , R . D . and Mor t on , W .,

" Di ffe rence Me thod s forIni t i al Value Prob lem s " , Inte r s c i enc e , New Y ork

, 1 967 .

Se r rin , J . ,

" Mathemat i c al Princ ip le s of C las s i c al F lui dMe chani c s Handbuch de r Phy s ik , Vol . 8

, 1 959 , 1 2 5 - 263 .

Smol le r , J . ,

"A uni quene s s the o rem for Riemann pr ob lem s

Ar c h . Rat . Me ch . Anal . , 33 , 1 969 , 1 1 0 - 1 1 5 .

S t rang , G . ,

"Ac c urate Par t i al Di ffe renc e Me thod s " , Num .

Math . , 6 , 1 9 64 , 3 7 - 46 .

Wend r off , B . ,

" The Riemann pr ob lem fo r mat e ri al s wi th nonc onvex e quat i on s of s t at e 1 1 ; gene ral fl ow”, J . Math . Anal .

Appl . , 3 8 , 1 97 2 , 64 0 - 65 8

Weyl , H . ,

" Shock wave s in arb i t rary f lui d s , C omm . Pu re Appl .

Math . , 2 , 1 94 8 , 1 03 - 1 2 2 .

Y anenko , N . N . ,

" The me tho d o f frac t i onal s t ep s ; the s olut i onof p rob lem s in mathemat i c al phy s i c s in s eve ral v ari ab le s "

E ngl i sh t rans lat i on , New Y ork , Sp ringe r- Ve rlag , 1 97 1 .

6 0

ON THE MATHE MATICAL THE ORY OF DE FLAGRATIONS AND DE TONATIONS

K . O . Fr ie dr i chs

C ourant Ins t i tut e of Mathemat i c al Sc i enc e sNew York Un ive r s i ty

New York , New Y ork 1 0 0 1 2

Whi le the p r opagat i on of a s hock wave i s c omp le t e ly de t e r

mine d by the c ons e rvat i on laws , the b oundary c ondit i ons of the

p r ob lem , and the addi t i onal c ond i t i on that the ent r opy inc re as e in

t he p roc e s s , the s ame i s not t rue fo r the p r opagat i on of a de t ona

ti on wave and of the flame f ront in an ordinary c ombus t i on pr oc e s s .

More c ondi t i ons mus t b e adde d t o the c ons e rvat i on laws in or de r t o

pr ovi de s uffi c i ent data for the uni que de te rminat i on of the p ropa

ga tion pr oc e s s . Fo r de t onat i on s thi s ne c e s s i ty was re c ogni z e d by

Chapman and Jougue t when t hey int roduc e d t he i r famous hyp othe s i s .

For c ombus t i on pr oc e s s e s thi s ne c e s s i ty wa s more or le s s t ac i t ly

as sume d by Jougue t and othe r s when they at t acke d the c alc ulat i on

of the flame spe e d by t ak ing he at c onduc t i on int o ac c ount wi thout

eve n t ry ing t o de te rmine the flame s pe e d f rom the c ons e rvat ion laws

and b oundary c ond i t i ons alone .

It is natural to expe c t that the nee de d add i t i onal c ondi t i ons

c ould be de rive d f rom an inve s t igat i on of the inte rnal me chani sm

of the c ombus t i on o r de t onat i on p roc e s s . Thus v . Neumann [ 1 ] has

arrived at a j us t i fi c at i on o f the Chapman- Jougue t hypo the s i s fo r

de t onat i ons by t aking int o ac c oun t that the chemi c al re ac t i on take s

place ove r a z one of fini t e wid th ; hi s argument s are bas e d on the

as s ump ti on that a de t onat i on i s ini t i ate d by a shock . For combus

6 1

t i on pr oc e s s e s , whi ch do not involve s hocks,no un i que de t e rmina

t i on c an be achi eve d wi thout t aking he at c onduc t i on int o ac c oun t .

It is the int ent i on of the p re s ent pape r t o of fe r a uni fi e d

and more c omp le te d i s cus s i on of the que s t i on of de te rm inacy for

de t onat i ons and de fl agra t ionsl

. In or de r t o b e ab le t o p oint out

the c ont ras t b e twee n the s e two kind s of p r oce s s e s we shall tre at

b o th of them on the b as i s of the s ame as s ump t i ons : We shall t ake

v i s c o s i ty and he at c onduc t i on int o ac c ount and as s ume a fini t e rat e

o f chemi c al r eac t i on . ( Ac c o rdingly , we shall not p o s tulate that a

de t onat i on p roc e s s b egins wi th a s hock . ) Fr om the d i s cus s i on of

the int e rnal me chani sm of the de t onat i on and de flag rat i on p roc e s s

on thi s b as i s we shall ob t ain the de s i re d add i t i onal c ond i t i ons

whi c h make uni que de t e rm inat i on of the whole p r oc e s s p o s s ib le by

exc luding c e rt ain de t onat i on or de flagrat i on p roc e s s e s whi ch would

b e c ompat ib le wi th the c ons e rvat i on laws . In par t i cular we shall

find as a re s ul t that a de t onat i on beg ins wi th a s hock and that the

Chapman- Jougue t hyp othe s i s furni she s the c o rre c t addi t i onal c ond i

t i on p r ovi de d that for a given value of the re ac t i on rate the vi s

c osity and the heat c onduc t ivi ty are s uff i c i ent ly small . I f , on

the o the r hand , the re ac t i on rate i s ve ry high , for g iven vi s c os i ty

and he at c onduc t ivi ty , the de t onat i on no l onge r b egins wi th a

s hock ; and i f the re ac t i on rate is exc e s s ive ly high , the Chapman

We p ropo s e t o u s e the t e rm de flagrat i on for tho s e c ombus t i onpr oc e s s e s whi ch t ake plac e in a ve ry nar row z one of c ons t ant wi dthand whi c h the re f or e in good appr oximat i on c an be de s c rib e d by adi s c ont inui ty . For de t onat i on and de flag rat i on p roc e s s e s we s hallemp loy the c omm on name " re ac t i on p r oc e s s "

62

https://www.forgottenbooks.com/join

quant i t i e s ins i de the re ac t i on z one and , s e c ondly , that the t ime

rate of change of the wi dth of the re ac t i on z one i s small when

c ompare d wi th the ave rage s pe ed wi th whi c h the ga s e s c r os s the

re ac t i on z one . Thi s a s sump t i on appe ar s t o b e j us t i fi e d only i f the

c oe ffi c i ent s of vi s c o s i ty and he at c onduc t i on are s uffi c ient ly

small and the rate of re ac t i on is suffi c ient ly large . ( We le ave

as i de the que s t i on whe the r or not unde r the s e c i r cum s t ance s the

as sumpt i on i s always j us t i fi e d . ) The pe rt inent quant i t i e s on b oth

s ide s of the re ac t i on fr ont " are then c onne c te d by the s ame we l l

known laws of c ons e rvat i on of mas s , momentum , and e ne rgy that hold

for the quant i t i e s at b oth s i de s of a di s c ont inui ty s urface .

Next we g ive a b r i e f ac c ount of the inde t e rminac ie s that one

enc ount e rs when one t ri e s t o de t e rm ine a flow involv ing a re ac t i on

d i s c ont inui ty s o le ly by us ing the c ons e rvat i on laws and the bound

ary c ondi t i ons . To thi s end i t i s ne c e s s ary t o di s t ingui sh vari ous

type s of re ac t i on p roc e s s e s . Among the de t onat i ons the re is , as i s

we ll known , a part i cular one , the " Chapman- Jougue t" de t onat i on ,

whi c h i s s ingle d out f rom othe r s by the p r ope r ty that the flow of

the burnt gas i s s oni c when ob s e rve d f rom the re ac t i on front . We

have te rme d s t r ong” or ”we ak" a de t onat i on i f i t involve s a

pre s s ure ri s e g re ate r or le s s than fo r a Chapman- Jougue t de t ona

t ion .

lS im ilarly , we have te rme d s t rong" or " we ak " a de flagrat i on

i f i t involve s a pre s sure d r op gre ate r or le s s than for a Chapman

Jougue t de flagrat i on , whi ch again i s charac t e ri z e d by the c ond i t i on

that the flow burnt ga s i s s oni c when ob s e rve d fr om the re ac t i on

For the fo ll owing s e e

64

front . A c ons t ant volume de t onat i on i s the l imi t ing c as e of a

we ak one , pr oduc ing the le as t pre s sure ri s e ( and the le as t t empe ra

ture ri s e ) among al l de t onat i ons f or the s ame exp lo s ive ; a c ons tant

p re s s ure de flagrat i on i s al s o the limit ing c as e of a we ak one,

pr oduc ing the le as t dr op in dens i ty ( and the g re ate s t tempe rature

ri s e ) among all de flagrat i ons .

We now c ons i de r the f low of gas in a half- finit e tube re

su l ting , a s indi c ate d b e fore , when a re ac t i on f ront s t art s t o move

fr om the fini t e end of the tube int o the unburnt gas unde r the

influence of a p i s t on whi ch move s in a p re s c r ib e d manne r . We then

ask for the ga s mot i ons whi ch are c ompat ib le wi th the c ons e rvat i on

laws and the p i s t on mot i on . Mathemat i c ally spe aking , we a sk for

the s olut i ons of the flow d i ffe rent i al e quat i ons c ompat ib le with

the t rans i t i on c ondi t i ons at the d i s c ont inui ty front and wi th the

b oundary c ondi t i on , whi c h exp re s s e s that the ga s ad j ac ent t o the

p i s t on has the s ame ve loc i ty as the p i s t on . Thi s p rob lem wi ll b e

re fe rre d t o a s the exte rnal f low p rob lem . The an swe r,

explaine d in de t ai l in [ 2 ] i s thi s :1

Supp o s e the re ac t i on

proc e s s is a de t onat i on ; i f then the p i s t on move s in the s ame

d i re c t i on a s the re ac t i on fr ont , and i f the ve loc i ty uP

of the

pi s t on exc ee d s the ga s ve loc i ty u whi ch would be p roduc e d by aD

Chapman- Jougue t de t onat i on , then the re is j us t one flow involving

I t s hould be emphas i z e d that the the o ry as c ons ide re d in thi sre port , bas e d on the as sumpt i on that the re ac ti on front be a sharpd i s c ontinui ty , d oe s not offe r any po s s ib i li ty of pre di c t ing whe the ra de t onat i on or a de flag rat i on wi ll oc c ur in a given s i tuat i on( exc e p t that unde r c e rt ain c i rcum s t anc e s de flag rat i on flow i s no tp os s ib le ) .

p i s t on ve loc i ty . If , howeve r , the p i s t on ve loc i ty uPis le s s than

u ad j us tment of the ve loc i ty of the burnt gas t o the p i s t onD’

ve loc i ty c an alway s b e ac hi eve d by a Cha pman— Jougue t de t onat i on

fol l owe d by an app ropr i at e rare fac t i on wave , but i t c an al s o b e

ac hi eve d by a s e t of we ak de t onat i ons fol lowe d e i the r by a sh ock

or a rare fac t i on wave . Thus , in c as e uP

uD, the s olut i on of the

mathemat i c al pr oblem i s not uni que ; the re i s a one - parame t ri c set

of s o lut i ons .

s t i l l h ighe r . To any p i s t on ve lo c i ty uP

one ha s s t i ll the choi c e

of a flame ve l oc i ty arb i t rary wi thin c e r t ain l im i t s1

and c an

achi eve ad j us tment of the gas ve loc i ty t o the p i s t on ve loc i ty by

s end ing ahe ad of the flame a sho ck of appr op r iat e s t rength . Again ,

the re i s a one - parame t ri c s e t of s olut i ons as long as the de flagra

one s , the po s s ib i l i t ie s of ad j us tment b e c ome s t i l l great e r and the

s e t of s olut ions i s two— parame t ri c .

To expr e s s the s e de t e rminacy s t atement s in a s imp le f orm we

int r oduc e as " degre e of unde r- de t e rminacy " of the exte rnal flow

pr ob lem the numbe r t of c ond i t i on s that mus t b e imp o s e d on the data

of the flow pr ob lem in orde r t o make the s olut i on uni que .

Summar i z ing we then have

The l im i t ing c as e i s the one in whi ch the flame t oge the r Wi ththe pre — c ompre s s i on sho ck are j us t e quivalent t o a de t onat ion .

66

S t rong de t onat i ons t O

We ak de t onat i ons t 1

We ak de f lag rat i ons t l

S t rong de f lagrat i ons t 2

Chapman- Jougu e t de t onat i ons and de f lagrat i ons are he re c las s e d wi th

s t rong de t onat i ons or we ak de flagrat i ons re s pe c t ive ly .

The s e pe c ul i ar unde r- de t e rm inac i e s are a c ons e quenc e of

Jougue t's imp or t ant rule ( c f . c onc e rning the pr ope r t i e s of

the ga s f low ob s e rved from the re ac t i on fr ont

W eak detonatlon Strong detonation

W eakdeflagratron. Strong deflagration.

Part i c le paths and sound paths /f

a t

de t onati on o r de flagrat i on front s .

is he re alway s unde r s t ood re lat ive t o the re ac t i on fr ont .

We now p r oc e e d t o d i s c u s s the c hi e f aim of the p re s en t pape r ,

name ly t o de c i de whi c h of the f low pr oc e s s e s , s t i l l pe rm i t t e d by

the c ons e rvat i on laws , are exc lude d through the ac t i on o f vi s c o s i ty

he at c onduc t i on , and chemi c al re ac t i on . To thi s end we shall t ake

int o ac c ount that the re ac t i on z one has a fini t e ext ens i on . We

then s hall s e t up the d i ffe rent i al e quat i ons g ove rning the t ran s i

t i on ac ro s s such a re ac t i on z one and inve s t i gat e unde r whi c h c i r

c ums t anc e s the s e d i f fe rent i al e quat i ons po s s e s s s olut i ons s at i s

fying the b oundary c ond i t i on s imp o s e d at the two end s of the re

ac t i on z one . The s e b oundary c ondi t i on s c ons i s t in pre s c r ib ing the

chemi c al c ompo s i t i on , p re s s ure , tempe rature , and ve loc i ty of the

gas e s at b o th end s of the re ac t i on z one in such a way that the laws

of c ons e rvat i on of mas s,momentum , and ene rgy are s at i s fi e d by

the s e quant i t i e s . The d i ffe rent i al e quat i ons in the int e r i or of

the re ac t i on z one exp re s s the same c ons e rvat i on law s , but t ake int o

ac c ount chemi c al reac t i on, vi s c os i ty , and he at c onduc t i on .

Inve s t igat i ons of thi s kind for shock s not involving a

chemi c al re ac t i on have b e en made in de t ai l by vari ous author s1

The re sult wa s that a t rans i t i on b e twe en the given quant i t i e s at

b oth s ide s of the z one is alway s p o s s ib le p r ovi de d that the

di re c t i on of the flow c or re s p ond s t o inc re as ing ent r opy . Our

re ac t i on , howeve r , the s i tuat i on i s c omple te ly d i ffe rent . He re are

the re sul t s of our analy s i s

Unle s s the rate of re ac t i on is exc e s s ive ly high , we ak de t ona

The c ondi t i on that the de t onat i on b e s t rong or of the Chapman

Jougue t type is , the re f ore , the de s i re d add i t i onal c ond i t i on men

tioned in the be ginning . C onse quent ly , a f low invo lv ing a de t ona

t i on is uni que ly de te rm ine d . Fo r , i f the p i s t on ve lo c i ty i s high ,

u u the re i s a uni que s olut i on involving a s t rong de t onat i on ,P D’

as ment i one d be f ore ; for le s s e r p i s t on ve loc i ty , howeve r , uP .: u

D,

the r e is now only one po s s ib le flow le ft in which the ve loc i ty of

the burne d ga s e qual s the p i s t on ve loc i ty , and that i s the flow

inv olving a Chapman- Jouguet de t onat i on . In part i c ular we see that

fo r a tube Wi th a c l o s e d end , uP

O , or open end , uP

0 , a

oc c urrenc e of thi s par t i c ular de t onat i on i s he re de duc e d and no

add i t i onal hypo the s i s is re qui re d .

If , howeve r , the re ac t i on rat e i s exc e s s ive ly high , the

analy s i s y i e ld s the re s ult that the Chapman - Jougue t de t onat i on i s

impo s s ib le . Ins t e ad , a par t i c ular we ak de t onat i on i s pos s ible

whi c h t rave ls wi th a we ll - de te rmine d ve loc i ty ( de pend ing on

pre s s ure and tempe rature in the unburnt gas , re ac t i on rat e , vi s c os

i ty , and he at c onduc t ivi ty ) . I f the pi s t on ve loc i ty i s large

enough , ad j us tment s t i ll re qui re s a s t r ong de tonat i on . For le s s e r

69

p i s t on ve loc i t i e s , for examp le for c lo s e d or open end s , ad j us tment

i s e ffe c te d by the part i c ular we ak de t onat i on fo llowe d by an

app r opr i ate shock or an appr opr i ate rare fac t i on wave .

As to de f lagrat i ons the re s ult s o f the analys i s are that

the f lame ve loc i ty , i . e . the ve loc i ty of the re ac t i on fr ont

re lat ive t o the tube , fr om the I‘bu rning s pe e d , i . e . the s pe e d of

the re ac t i on front re lat ive t o the unburnt gas ahe ad of i t . Whi le

the flame ve loc i ty de pend s on the b oundary c ond i t i ons of the

ular value i s the de s i re d addi t i onal c ond i t i on for de f lagrat i ons .

A s s tat e d b e fore , fo r a burning s pe e d arb i t r ary wi thin c e r t ain

l im i t s , a de f lagrat i on flow c an be found whi c h i s adapte d t o the

p i s t on mot i on . A furthe r l imi t at ion i s imp o s e d by exc lud ing s t rong

de flag rat i ons . Thus we s e e : Wi thin c e rt ain l imi t s f or the dat a of

de flagrat i on and adap te d t o the p i s t on mot i on .

We c all at t ent i on t o a numbe r of de t ai le d inve s t igat i on s of

the t rans i t ion proc e s s in re ac t i on z one s . De t onat i on t rans i t i on s

Thi s re s ult c ould als o b e e s t ab li she d by v . Neumann 's argum enti gnoring vi s c o s i ty and he at c onduc t i on and t aking only the fini t erat e of chem i c al re ac t i on int o ac c ount .

https://www.forgottenbooks.com/join

2 — 1 2

Thi s is in agre ement wi th our b as i c a s sumpt ion ( p . 3 ) that the

re ac t i on z one i s ve ry nar row and of ne ar ly c ons t ant wi dth . C on

s e quent ly , al l quant i t i e s de pend only on an ab s c i s s a x , and not on

the t ime . At e ach p lac e x , the re is a mixture of burnt and unburnt

ga s ; we deno te by e the frac t i on of mas s of burnt gas in the mix

ture . We denote p re s s ure and s pe c i fi c volume by p and T and int ro

duc e the re duc e d tempe rature 8 pT , a quant i ty whi c h has the

dimens i on of ve l oc i ty s quare d . We as s ume burnt and unburnt ga s t o

b e ide al . Ac c ord ingly , 8 i s p r op or t i onal t o the tempe r a t urel

.

The int e rnal ene rgy e pe r unit mas s of the burnt and of the unburnt

ga s i s as sume d t o b e a func t i on of 9 only , ( ac tual ly e de pend s al s o

s omewhat on the p re s s ure p , s e e al s o footnot el: deno t ing by g the

ene rgy of format i on pe r uni t mas s ( at ab s olute z e r o t empe rature ) ,

we int roduc e the t ot al ene rgy pe r uni t mas s

E e +-

g

Burnt and unburnt gas are d i s t ingui she d by the supe r s c r ip t s

( O ) . The t ot al ene rgy pe r uni t mas s of the mixt ure i s then

1( 1

On the ene rgy func t i ons E ( 8 ) we re qui re that

The ab s o lute t empe rature i s given by Re/M whe re R i s the gasc ons t ant and M the mole c ular we ight . We di s re gard the de pendenc eof the mole c ular we ight on the m ixture rat i o 2 .

7 2

Thi s c ond i t i on implie s that the " libe rate d ene rgy

i s po s i t ive . ( It i s c onveni ent t o make thi s as sump t ion although

mos t of our c onc lus i ons hold wi thout i t . ) Our re qui rement is

s at i s fi e d i f burnt and unburnt ga s are poly t rop i c " ; i . e . i f the

ene rgi e s are given by

( 0 ) 8 l eE E (

wi th c ons t ant yo, yl

, go, gl

, pr ovided that the tempe rature is

b e low a c e r tain lim i t

B y v we denote the ve loc i ty of the s t e ady gas f low and by

m T 1v the mas s f lux" of m ixture thr ough a unit c r os s - s e c t i on

pe r uni t t ime . B y S we deno te the mas s of burnt gas c re at e d pe r

uni t mas s of unburnt gas pe r uni t t ime . We as sume that the

" re ac t i on rat e , S , de pends on 9 and p , ( c f . foo tnot e

and that S vani she s be low a c e rt ain " s afe ty t empe rature

S O for 8 es

2

( Y OThi s limi t c orre s pond s t o 9 ( g g I f one

a s sume s yo 7 1

m the mole cular we ight Mo

M1

m 3 0 , and

the " libe rate d ene rgy "

gO

- glm . 7 kc a l/gr , then the l im i t i s about

3 2 OOOK . Thi s c as e i s , howeve r , unre al i s t i c s inc e f or such tempe ra

ture s the value of y for c ombus t ib le gas e s wi ll hard ly eve r be ashigh as

2The lat t e r as s umpt i on i s made oh l t o achi eve mathemat i cal s im

plic ity . In re c ent pape r s [ 1 0 the reac t i on rate i sas s ume d t o be of the form

( foo tnote c ont inue d )

73

The t rans i t i on b e twe en the two s t at e s on b oth s i de s of the

re ac t i on fr ont i s e ffe c t e d by the ac t i on of vi s c os i ty and he at

c onduc tionl

. We int roduc e c or re s ponding c oe f fi c i ent s u and A such

that

dy de“ as

and “a

are the vi s c ous pre s sure and he at pe r uni t mas s c onduc t e d th rough

a uni t c r os s - s e c t i on pe r uni t timeg

. The s e c oe f fi c i ent s de pend on

p,T , and a ; but we nee d pay only l i t t le at t ent i on t o thi s de pen

dence of our gene ral d i s cus s i on .

We now fo rmulate the laws gove rning the p ro c e s s . The c ont i

nuity e quat i on s imp ly as sume s the f orm

( fo otno te c ontinued )

S S e' A/G

whe re A , pr opor t i onal t o the ac t ivat i on ene rgy , i s s o large that Sis negligib le when 9 as sume s value s c orre sponding t o a t empe ratureof 3 OO

OK . The re duc e d s afe ty t empe rature G

Shas no p re c i s e signif

ic anc e ( as the igni t i on t empe rature ha s in the olde r l i te rature ) ;i t i s s imp ly a value b e l ow whi c h S can be s e t e qual t o z e r o for allprac t i c al purp o s e s . The maximal re ac t i on r ate , S

G D’ may depend

on P .

We ignore d i ffus i on and rad iat i on . Di ffus i on should not b enegle c t e d in the ac tual c alculat i on of flame s pe e d s ac c ording t o

We fe lt that fo r the s ake of s impl i c i ty we c ould di s regarddi ffus i on s inc e the addi t i onal t e rm s due t o i t would not s e em t oent ai l any e s s ent i al change in the ge ne ral re s ult s .

One might ob j e c t t o us ing the not i ons vi s c o s i ty , he at c onduc

t i on , and d i ffus i on i f the wi dth of the t rans i t i on z one i s ext remely smal l . I t s e em s l ike ly , though

,that neve rthe le s s our

re sult s remain c or re c t in quali t at ive re s pe c t s , in par t i cular a s

far as de t e rm inacy i s c onc e rne d .

2The cus t omary c oe ffi c i ent s of vi s c os i ty and heat c onduc t i on are ,

in our not at i on, 3 /4u and A/R , R be ing the ga s c ons t ant such that

R9 i s the t empe rature .

( O ) m c ons t .

The l aw of c ons e rvat i on of momentum i s

I( l ) udx

-r p-rmv P c ons t .

C ons e rvat i on of ene rgy 1 i s exp re s s e d by

de ( e ) l 2 dv( 2 ) A

ai-t m[ E 2

v -rv [ p u dx ] mQ c ons t .

The b alanc e b e twe en burnt and unburnt gas is given by

or -v o ,

2as suming a fi r s t orde r re ac t i on fr om a unim ole c ular me chani sm

The p rob lem i s t o inve s t igat e po s s ib le s olut i ons of the s e

d i ffe rent i al e quat i ons i f the value s of the quant i t i e s v , p , T ,

and s are given at the end po int s of the re ac t i on z one . We mod i fy

thi s p robl em by pre s c r ib ing the s ame value s of the s e quant i t i e s at

x a ) and x - 0 0 . That i t i s j us t i fie d wi th good appr oximat i on

t o s ub s t i t ut e the modi fi e d prob lem fo r the orig inal one fol lows

If we we re t o c ons ide r d i f fus i on we would int roduc e a c oe ffi c ient6 such that - 6 de/dx i s the frac t i on o f mas s of burnt gas d i ffus ingthrough a c r o s s - s e c t i on pe r un i t t ime . Then we would have t o addthe t e rm Td ( 6 d e/dx ) dx t o e quat i on and -t

§- v ] /dx

to e quat i on in orde r t o exp re s s the d i ffus i on of ene rgy . There s ul t ing mod i fi e d e quat i on would d i ffe r s omewhat f rom tho s ein the li te rature ( s ee [ 1 1 ] whe re only the d i ffus i on of the ene rgyof fo rmat ion i s t aken int o ac c oun t .

2If a

(diff

grent me chani sm of re ac t i on we re as sume d le ading t o

te rm s 1 E; or ( l c f . [ 1 0 ] and no change in thegene ral c onc lus i on s would re s ul t .

75

fr om our b as i c as s ump t i on ( p . 3 ) that the wi dth of the re ac t i on

z one is ve ry nar row . More s pe c i fi c ally , the as s umpt i on was that

the rat e of change o f the pe rt inent quant i t i e s out s i de the re ac t i on

z one is negligib ly small a s c ompare d t o the rat e s of change o f

the s e quant i t ie s ins i de the re ac t i on z one . C ons e quent ly,

the s e

quant i t i e s appe ar t o be ne ar ly c ons tant at the end s of the re ac t i on

z one ove r a re gi on who s e ext ens i on i s large c ompare d wi th the wi dt h

of the re ac t i on z one . I t i s then natural t o as sume that the

proc e s s ins i de the re ac t i on z one c an ve ry we ll b e appr oximate d by a

p ro c e s s that extend s ove r the whole fie l d f rom x - 0 3 to x + a >

and in whi c h the pe r t inent quant i t i e s as sume at t oo tho s e value s

that are p re s c r ibe d f or the p rope r p r oc e s s at the end s of the

fini t e re ac t i on z one .

Ac c ord ingly we ask for s olut i ons of the e quat i ons

and whi c h are de f ine d for - d a x 0 0 and whi ch app roach

fini t e lim i t value s ( wi th T O ) as x -e - i a a ; then the de rivat ive s

appr oach z e r o as s e en fr om and S olut i ons whi c h

b ehave in that way at x a ) or x - 0 0 wi ll be c alle d " re gular"

Thi s pro c e dure , typ i c a l f o r the t re atment of " b oundary laye rphenomena" is alway s emp loye d fo r d if fe rent i al e quat i ons in whi chthe t e rms of highe s t o rde r are mult ip li e d by small fac t o rs . ( In

our c a s e the s e fac t o rs ar e u , A , and Sag. )

No ac c uracy would be gaine d by t ry ing t o d i s c us s the s olut i onsfor a fini t e range x x x for , the c ons e rvat i on law s ( l )

'

o 1 land ( 2 )d> would not b e ac curat e ly val i d unle s s ac c i dentally

d t de d e

3 ?-

d—

X fi0 at X = X

Oand X = x

1

F o r the infinit e range i t foll ows fr om regular i ty ( s e e Se c t i on 2 )t hat the s e de rivat ive s vani sh at the end po int s .

the re . The lim it value s for x - d > are denote d by po, T

o,

0

and so; tho s e for x -

a > by pl, T

1, 8

1, and 8

1' The b oundary

c ondi t i ons then c ons i s t in p re s c r ib ing the s e value s ; in part i cular

we p re s c r ib e

exp re s s ing that the ga s c ons i s t s of unburnt ga s at x - d > and of

c omple te ly burnt gas at x + d 3 . From re lat i on we then de duc e

that the flux m is po s i t ive

m m d c ( T— l ) ( l 0

From thi s fac t i t foll ow s that the re ac t i on begins in the unburnt

gas at x - a > and end s in the burnt gas at x a) .

l ’1’ v

1for unburnt and

burnt ga s are tho s e that are pre s c r ib e d at b oth s i de s of the d i s

66The value s po, v

0, and p

l, IT

o’

0,

c ont inui ty front . The s e quant i t i e s are not pre s c rib e d arb it rar i ly ,

they are t o s at i s fy the c ons e rvat i on laws : ( 0 ) and

t

( l ) po+ mv

Op l

i- mvl

P

l 2 l 2 _( 2 )

thr ough whi c h at the s ame t ime the value s of the c ons tant s P and Q

are de te rm ined . He re we have se t

E0

and E( 1 )( 8o ll )

The c ons e rvat i on laws fo llow imme di at e ly for any regular

s olut i on from the d i ffe rent i al e quat ions and s inc e the se

77

2 — 1 8

e quat i ons re duc e t o ( l )' and ( 2 )oo fo r x a ) and x - oa

A furthe r as s umpt i on whi ch we imp os e on our b oundary value s

expre s s ing that no re ac t i on c an t ake p lac e in the unburnt gas at

i t s ini t i al t empe rature , and that re ac t i on would take p lac e at the

final tempe rature i f unburnt gas we re s t i ll le ft .

1

F or the foll owing argument s i t i s c onvenient t o e liminat e v

and p by

v mr and p T

- le

and t o c ons i de r T , 9 and e as the only de pende nt variab le s ; the

e quat i on s then be c ome

( 1 ) - um -

a§+ r 9 + m T = P ,

( 2 ) A m212+ 2 1 ] mQ

( 3 ) 0

The c ons tant c oe ffi c ient s m , P, Q , and the b oundary value s TO, T

1,

80, 8

1, s

oO , and 8

1l are s ub j e c t t o the c ons e rvat i on law s

1 2 l 2( 1 )

OT 9

-t m To

Tlel

-+ m Tl

P

l 2 2 l 2 2( 2 )

o

Re lat i on 80

88would imply S O fo r 8 8

0and would henc e not

b e c omp at ib le wi th e quat i on fo r a re gular s olut i on .

7 8

https://www.forgottenbooks.com/join

e quat i ons . The be hav i or of the rlparame t ri c s et of regul ar s olu

t i ons c an b e charac t e r i z e d by the re gular s olut i ons of the

line ari z e d e quat i ons ; the lat t e r are a l ine a r c omb inat i on of expo

nentia l func t i ons eQLX

i f the charac t e r i s t i c r oot s are d i ffe rent ;

o the rwi s e t e rm s like eLX

or xgemX

ente r . In c as e one charac t e r

istic exp onent is z e r o , one c annot s ay o ff - hand what the de gree of

regular i ty i s ; a spe c i al c ons ide rat i on i s nee de d .

The lineari z e d e quat i ons at x a ) for the quant i t i e s T- Tl ,

8 - 91 , s — l are imme d iate ly foun d t o be

d ( T— Tl _2 2 - l

mil l T+ T

i91( T ' T

l) “ m ( T ' T

l )“1 ( M i

l ) 0

d ( e- 8 1 ) 8 — 81 _1

xl T

— m

F— AE

1 ( €- l ) — T

1G1 (

T- ’

rl )

O

d ( e— l ) 1mT

+ TlSl( e- l ) O

whe re we have s e t as E( O )

- E( l ) and the sub

y- l T ’

s c r i pt ( 1 ) indi c at e s that the s e quant i t i e s and al s o u , A , and S

are t o b e t aken fo r e 91, T T

1 , 2 1 .

F or the charac te r i s t i c exp onent a one then ob tains the e qua

t i on

8 0

2 — 2 1

2 2 2 2[M i

a “11 1fii‘l

' 91h

- m2T2

9 me

O1 vi

”Y i

’yi 1

” T1

The fi r s t b racke t has evi dent ly one negat ive root . As t o the

s e c ond b racke t , whi ch we wr i t e in the fo rm a a2

- b a -r c we ob s e rve

that i t ha s re al r oot s s inc e the di s c riminant

2 2

2 2 2 2 2 LL1 2

m Tlxlul 2 2

b — ua c ( m TlAl

GlAl

-i- m T1 F

) 4

T( Vlel

m T1 )

2 2 2 2 LLl 2 2 2

(m TlAl

- 81A1

- m Tl TIrI

) 4m TlAlul

el

i s pos i t ive .

S inc e a 0 we s e e that the s e c ond b racke t ha s one po s i t ive

and one negat ive root i f c 0 . I f c 0 , we have

7 1b b

O( yl Olu l O and henc e the b racke t has one

vani shing and one p os i t ive ro ot . I f c 0 we have b bO

O and

henc e the b racke t ha s two po s i t ive roo t s .

In c as e c 0 we have r 2 and in c as e c 0 we have rl==l .

l

A de t ai le d inve s t igat i on of the ve c t or fi e ld c or re s ponding t o the

di f fe rent i al e quat i on would s how that in c as e c O a two parame t ri

s e t of regular s olut i ons exi s t s ; hence r2

2 al s o in thi s c as e .

S inc e the c ond i t i on c 2 O i s e quivalent wi th ylel2 m

eTiwe have t o

d i s t ingui sh the following tw o c as e s

Cas e ( A1 )m T

Ca s e ( B 1 ) m T

The degre e of re gulari ty in t he s e c as e s is

C as e ( Al )

II R)C as e ( B r

1) 1

E mploying the s ound s pe e d cl

/y 191

fo r

s t at e ( 1 ) we c an wr i t e the c ondi t i on for c a s e s

the form

C as e ( Al )

C as e ( B l )

Thus the flow of the burnt gas in s t at e ( 1 ) i s s upe r s oni c in

c as e ( A1 ) and s ub s oni c o r s oni c in c as e ( B 1 )°

For the s t at e ( 0 ) at the end x - 0 3 we ob t ain a s imi lar

e quat i on fo r a , the only di ffe renc e b e ing that S O in s t at e ( 0 )

s inc e 80

8 wa s a s sume d . The e quat i on for a then be c ome sS

ma T2A a

2- (m

2T2A - 9 A + m

2T2 n

o) ao o

uo o o o o o y

O

- I

m 2 2—

_L' ( Y 9 T O

y oO O 0

whe re AO, u

o, yo

re fe r t o 8 90, T T

o, 2 O . The fi r s t fac t or

he re has the root 0 O . The b racke t alway s has one p os i t ive root

and in ad di t i on one po s i t ive , vani shing , or negat ive roo t de pend ing

2 2on whe the r y o

GO

- m T O , O , or 0 .

To de t e rm ine the de gre e of regulari ty rO

at the end x - d >

we fi r s t re c al l that 80

88was as sume d . Henc e for eve ry regular

s olut i on as suming the p re s c r ib e d b oundary value s at x - o> we

8 2

have 8 as

i f x is suffi c i ent ly negat ive, for - d > x x s ay .

3

O fo rBy vi r tue of S O for 8 ase quat i on ( 3 ) ent ai l s e

- G ) x xs

. The inve s t igat i on of the mani fold of s olut i ons

re gular at x - oa i s thus re duc e d t o the inve s t igat i on of the

regular s olut i ons of e quat i on ( 1 ) and ( 2 ) wi th

The charac te ri s t i c exponent s f or thi s prob lem are the roo t s of the

b racke t in re lat i on Hence we c onc lude : If b oth root s of the

b racke t are p os i t ive we have r0

2 . The s ame is t rue i f one of

the r oot s is z e r o , a s a de t ai le d inve s t igat i on of the ve c t o r fi e ld

c orre sp onding t o the di ffe rent i al e quat i on would show . If one of

the r oot s is negat ive , howeve r , the o the r one be ing pos i t ive , we

have r1

1 . Ac c or ding ly , we d i s t ingui sh the foll owing two c as e s

Ca s e ( AO) m

gTO

R)

2 2C as e ( BO ) m T

o7 080

The degree of re gulari ty is

in C as e ( AO)

0in C as e ( BO )

E mploy ing the s ound s pee d cO

in the unburnt gas we wri te the

c ond i t i on fo r c as e s ( AO) and ( D

O) in the form

C as e ( AO) v

0 1co

Cas e ( Bo) v

0co

Thus , the flow of the unburnt gas at x - a 3 i s supe rsoni c or sonic

the re in c as e AO, whi le i t i s sub soni c in c ase B

0

8 3

2 - 24

We s ee that four di ffe rent c as e s are t o b e di s t ingui she d

ac c ording t o whe the r c as e A or c as e B ob t ains at x a ) or at

rule ( s e e p . 2 1 5 ) the s e four c as e s j us t c or re sp ond t o the

four c as e s of s t rong and we ak de t onat i ons and de flagrat i ons ;

the Chapman- Jougue t de t onat i on o r de flagrat i on , charac te r i z e d by

V c is he re c las s e d wi th the s t rong de t onat i ons or we ak1 l ’

de f lagrat i ons re s pe c t ive ly .

We now shall de te rmine in a fo rmal way , s imp ly by c ount ing

the numbe r o f parame te r s , the mani fold of s olut i ons whi ch are re gu

lar at b oth endpoint s . The s e t of s olut i ons whi ch are regular at

x o) i s rl- p arame t ri c . Am ong the s e s olut i ons tho s e are t o be

s e le c t e d whi ch are regular at x — 0 3 . S ince al l s olut i ons regular

at x - d ) f orm a rO

- parame t ri c s e t in the three - parame t r i c s e t of

all s olut i ons , i t i s c le ar that the c ond i t i on t o b e re gular at

x - d > i s expre s s e d by 3 - rO

re lat i ons . Thus 3 - rO

c ondi t i ons are

impo s e d on the r parame te r s charac t e r i z ing the s olut i ons re gular1

at x 0 0 . One of the s e parame te r s c an alway s b e c ho s en arb i t rar

i ly ( wi thin l im it s ) , s inc e f rom eve ry s olut i on s at i s fy ing the

b oundary c ondi t i ons one ob t ains a s e t of othe r s by sub s t i tut ing

x i— c ons t . for x . Thus 3 - rO

c ond i t i ons are imp o s e d on rl- l parame

te r s . If rl- l 3 - r an ( r

od- r

l- 4 ) - parame t r i c s e t of s olut i ons

0

c an be expe c te d t o e xi s t . If rl- l 3 - r

o, one s olut i on ( or e l s e a

fini t e numb e r of them ) c an be expe c t e d t o exi s t . I f rl- l 3 - r

O

more c ond i t i ons are impos e d than parame te r s are avai lab le . The s e

c ondi t i ons wi ll b e s at i s fi e d only i f the c oe f fi c i ent s ent e ring the

d i f fe rent i al e quat i ons or the b oundary value s as s ume app r opr i ate

8 4

value s . In othe r word s , 4 - rO

- rl

c ond i t i on s are imp o s e d on

c oe ffi c i ent s and b oundary value s . We te rm the numbe r

s 4 - rO

- rl

the " degree of ove r- de te rm inacy . From Jougue t's rule and the

de te rm inat i on of the value s of rO

and rl

given be fore we find

S t rong de t onat i on , Ca s e ( AOBl) s

We ak de t onat i on , C as e ( AoAl) s

We ak de flag rat i on , C as e ( BoBl )

s

S t rong de flagrat i on , C as e ( BoAl ) s

Upon c omparing thi s t ab le w ith the t ab le for the deg re e o f

unde r- de t e rminacy given in the Int r oduc t i on ( p . 7 ) we re ali z e the

c ond i t i ons ne e de d t o make the f low p rob lem uni que thus e qual s the

numb e r of c ond i t i ons impo s ed by the me c hani sm of the re ac t i on

pr oc e s s .

Thi s re sul t may be int e rpre te d a s f oll ows : Al l s t rong

de t onat i ons are po s s ib le . We ak or Chapman - Jougue t de t onat i ons are

only pos s ib le i f one of the parame te rs of the p roc e s s s at i s fi e s one

c ondi t i on . As s uch a pa rar ete r we may c ons i de r the flux m . As

we shall s e e late r , we ak de t onat i ons exi s t j us t for such value s of

the flux that le ad t o Chapman - Jougue t de t onat ions exc e pt for

suffi c ient ly high value s of the re ac t i on rate , fo r whi ch a large r

8 5

value of the flux le ad s t o a po s s ib le we ak de t onat i on . We ak

de flagrat i ons are als o only po s s ib le i f one of the parame t e r s,

the

flux m s ay , s at i s fie s one c ondi t i on . A s we shall s e e lat e r,weak

de f lagrat i ons exi s t indee d f or only a part i cular value of t he flux .

S t rong de flag rat i ons should exi s t only i f two c ond i t i ons are s at i s

f i e d by the parame te r s . As we s hall s e e late r , s t rong de flagra

t i ons d o not exi s t at all .

It mus t b e emphas i z e d that the s e s tat ement s are s o far

de rive d in a pure ly formal manne r . They are ob t aine d by b alanc ing

the numbe r o f avai lab le parame t e r s wi th the numb e r of c ond i t i ons

imp o s ed . A de fini t e s tatement ab out the exi s t ence and uni quene s s

c annot b e made on thi s b as i s . As a mat t e r of fac t the s e argument s

are not suffi c ient t o exc lude we ak de t onat i ons and s t r ong de flagra

t i ons . More de t ai le d c ons i de rat i on s are ne e de d for thi s purpo s e .

In the fol l owing Se c t i ons ( 3 and 4 ) we shall fi r s t inve s t igate

c e rt ain l imi t ing c as e s of we ak de f lagrat i ons and s t r ong de t onat i ons

and then p roc e e d t o di s c us s in Se c t i on 5 the p rob lem of exi s t enc e

of s olut i ons in ge ne ral .

The deg ree of ove r— de te rminacy was found t o be s 1 for

we ak de f lag rat i ons . As ind i c ate d ab ove , s uc h de flagrat i ons c an

the re fore b e expe c te d t o exi s t only i f the c oe ffi c i ent s o f the

d i ffe rent i al e quat i ons s at i s fy one c ond i t i on . Thi s c ondi t i on may

b e c ons i de re d a c ond i t i on for the c ons t ant m , the flux , or for the

burning ve lo c i ty vo

Tom . The que s t i on ari s e s whe the r or not t o

given value s of To, 6

0, and given func t i ons S ( 9 ) and the re

8 6

https://www.forgottenbooks.com/join

The e quat i ons ( 3 ) the n be c ome

_u m2dT/dx m2 . 15

—Ad6/dx 25

de/d'

i‘

t T - l ( 1 0

Thi s sy s t em o f d i f fe re nt i al e quat i ons dep ends on the parame

te rs m , A, u , P, Q, and 80

We ob tain the l im it ing p r ob lem by

c ons ide r ing a s e t of such sy s t ems for whi ch m app r oache s z e r o whi le

the o the r parame t e r s are ke p t fixe d . In othe r word s : The d i ffe r

ent ia l e quat i ons of the lim i t ing p r ob lem are s imply ob t aine d by

omi t t ing the three te rm s inv olv ing the fac t or mg

.

The fac t that A and u are fixe d whi le m appr oac he s z e ro

evi dent ly imp li e s ASOO/p

oo and “S

on/p

O0 . The l imi t ing e qua

t i ons wi ll the re f ore re pre s ent a good app rox imat i on i f ASOO/p

o,

“S a l /po ’ and m are small . In that c as e we c an re - int roduc e the

or iginal quant i t i e s . Thus we ob t ain the e quat i ons of the c ons tant

p re s sure p rob lem in the fo ll owing fo rm

( 1 ) p P pO

(2 ) - A %—i = mQ ,

(3 ) m ps' 1( 1 0

N

whe re we have e l im inat e d T f rom ( 2 ) and us ing

Re lat i on (I ) exp re s s e s the fac t that unde r the c ondi t i ons of

the limi t ing p rob lem the p re s s ure doe s not vary ac ro s s the flame

front : we have c ons tant p re s s ure c ombus t i on . Fur the r we s ee that

8 8

the te rm repre s ent ing the kine t i c ene rgy ha s dr oppe d out f rom

The b oundary c ond i t i ons are 8 81 ,

s l at x a ) ,and

l+ 8

1. The

re lat i on Bod- 8

0Q, whi ch hold s for regular s olut i ons de te rmine s

s O at x - 0 3 . The c ons t ant Q i s given by Q E

the value 8 80

at x - d > .

To inve s t igate p o s s ib le s olut i ons of thi s p rob lem one may

c ons i de r 8 as inde pendent vari ab le running from 80

t o 8 and c om1

b ine the e quat i ons t o

( 5 )

It is imme diate ly s e en that thi s e quat i on ha s a s addle - s ingulari ty

at the p oin t 8 81, a 1 . The re is j us t one s olut i on curve that

ent e r s thi s po int f rom the re gi on 2 l , 8 81

. If thi s curve i s

O at a point wi thfollowe d b ackwards i t wi l l ente r the axi s 6

9 9sprovide d p

oA/m

2i s suffi c i ent ly large ; i f p

oA/m

2i s suffi

c ient ly small the c urve wi l l ente r the l ine 8 88

at a point wi th

s 0 . It is thus c le ar that the re is j us t one value o f poA/m

2for

whi c h the curve ente r s the l ine 8 as

wi th s 0 . S inc e

d e/d8 O for 88, the curve wi ll ente r the line 8 8

0al s o

wi th the value a 0 . That for thi s s olut i on the quant i ty x

approache s a ) as 8 - al

and - d > as 8 80

i s imme d i ate ly s een from

and (3 )

Thus i t i s shown that in the l imi t , as the quanti t i e s

ASOD/p

oand uS

OO/p

oapproach z e ro , a de flag ra t i on p roc e s s exi s t s

wi th a we ll- de fined flux m and flame s pe ed vo

As wi ll be s hown

late r ( at the end of Se c tion the s ame i s t rue for small va lues

89

C omputat i ons of the burning spee d c an be c arri e d out by

s olving e quat i ons (2 ) and (3 ) or ( 5 ) thr ough inte rac t i ons , ( s e e

and in the lat t e r re p or t the t e rm d ( 6 d e/dx ) dx was

adde d t o e quat i on ( 3 ) in or de r to t ake di f fus i on int o ac c ount ) ; i t

was found in the quote d re por t s that for the ac tual s i tuat i ons

c ons i de re d the app roximat i on involve d in ( 2 ) and ( 3 ) was rathe r

ac curat e s inc e the omi t t e d t e rm s turne d out t o b e ve ry small for

the c alculate d s olut i on .

It i s int e re s t ing that e s s ent i al ly only the c omb inat i on

pA/m2

o r the d imens i onle s s c omb inat i on

( 6 ) eO/pAs

OD

ente r s the c ons t ant p re s s ure p rob l em , as s e en from e quat i on

(More pre c i s e ly , the p rob lem depend s only on the d imens i onle s s

quant i ty ( 6 ) in add i t i on t o the func t i ons S ( 8 )/S and

no te BO+ 8

OEli The quant i ty ( 6 ) appr oache s a fini t e l im i t

value a s ASOO/p

oappr oache s z e ro . The re fore , for small value s of

f—

fiASCD/p

Othe flux m i s p rop or t i onal t o p

OAS

OO80

and the burning

s pee d V0

t o ASOO8Opo

. If in par t i c ular A , Sq )

, and the re duc e d

ini t i al tempe rature 80

are ke p t fixe d , the flux inc r eas e s l ike

‘/SG D7P

Oand the burning s pe e d de c re as e s l ike ./S

oo7p

Oas p

O

inc re as e s . IfSOO we re inde pendent of p or inc re as e d wi th p

0of

le s s than fi r s t orde r,

the lat t e r re sult would mean that the

9 0

https://www.forgottenbooks.com/join

C ons i de r a s e quenc e o f s olut ions . The re are two p o s s ib i l idT d8

t i e s : e i the r the te rm s u a; and Aa?

in e quat i ons ( 2 ) d rop out in

the l im i t , or and be c ome infini te . For tho s e value s of x for

whi c h the f i r s t c as e oc c ur s , the e quat i ons

( 1 ) T 18 + m

21 P

( 2 ) —

2

1

( 3 ) -m 0 .dx

are s at i s fi e d in the l im i t . If the s e c ond c as e oc c ur s at a p lac e

x , a di s c ont inui ty of T and 8 o c cur s in the l imi t Whi le 2 remains

c ont inuous . Such a di s c ont inui ty would s imp ly b e a s ho ck not

involving a reac t i on .

A more de t ai le d inve s t igat i on ( s e e Se c t i on 5 ) of the l imit

pro c e s s ASG D/p

o, “Sa l /po

-fi 0 wi ll y ie ld that a s t rong de t onat i on

c an in the lim i t b e de s c r ib e d as a shock , not inv olving a reac t i on,

imme di at e ly fo l lowe d by a re ac t i on p r oc e s s g ove rne d by e quat i ons

( 1 ) ( 2 )

t i on p r oce s s , whi ch was f o rmul ate d m ore s pe c i fi c ally by G . I . Tay lo r

Thi s c onfi rm s the ac c ep te d i de a ab out a de t onaCD (I )

and v . Neumann ( s e e

The que s t i on ari s e s whethe r t o given value s of To, 8

0, T

1,

alc o m T1161

s at i s fy ing the c on s e rvat i on laws ,

a t rans i t i on p r oc e s s exi s t s whi c h c ons i s t s of a shock foll owe d by

81, and m wi th T

a re ac t i on . We as s ume that the t empe rature t o whi c h the shock

rai s e s the unburnt gas i s ab ove s afe ty t empe rature . Othe rwi s e the

reac t i on pr oc e s s would s imp ly b e a de flag rat i on pr oc e s s , whi c h

9 2

c annot exi s t in the p re s ent lim i t ing c as e l . Denot ing the quant i

t ie s pas t the shock front by an as t e ri sk we re qui re

To find out whethe r the e quat i ons ( 1 ) ( 2 ) ( 3 ) po s s e s s a s oluoo 0 3

t i on we de te rmine T and 8 a s func t i ons of s from ( 1 ) and ( 2 )oa

and ins e rt in Thi s i s p os s ib le i f the Jac ob ian

2 - 2 - 2

of ( l ) and ( 2 ) d oe s not vani sh . Int r oduc ing quant it i e s y

and O

( s )by

1 )

cit )we find

J (m2

- T - 1 )

whe re T,c , and y depend on 8 . S inc e the flow i s sub s oni c in the

s tat e pas t the shock we have J* O . In the ne ighb orhood of

the s tat e we may the re fore expre s s T and 8 in te rm s of e and

the s olut i on of the e quat i on re sult ing f rom ( 3 ) i s un i que ly de te r

mine d through the ini t i al c ond i t i on s O at x 0 , s ay , whe re we

may plac e the shock fr ont . The que s t i on then i s whethe r on c on

t inuing the s olut i on one would ob t ain a value for whi c h J c hange s

E xpre s s ing T and 8 through 8 by ( l ) and equat i on ( 3 )

b e c ome s a d i f fe rent i al e quat i on for e , whi c h yie ld s de/dx O for

e 0 s inc e S ( eo) O fo r 8

0 i85

. The s ole soluti on of thi sd i ffe rent i al e quat ion vani shing for x - 0 0 i s the refore s 0 .

s ign . The final s t ate i s al s o s ub s oni c , Jl

O , and the t rans i

t i ons f rom the s t at e to any o f the inte rme d i at e s t at e s b e twe en

and ( 1 ) c or re s pond , a s regard s the c ons e rvat i on l aws,

t o a s et

of we ak de fl ag rat i ons . All the inte rme d i at e s t at e s are thus sub

s oni c . C ons e quent ly , J remains negat ive thr oughout

The di f fe rent i al e quat i on for a re s ul t ing from ( 3 ) p o s s e s s e s ,

the re f or e , a s olut i on wi th e O fo r x 0 . S ince for thi s s olu

t i on , d s/dx -e - O as s l , the final s t at e ( 1 ) i s app r oac he d as

x a ) . Thus i t is s e en that a l im i t ing type de t onat i on c ons i s t ing

of a s hock foll owe d by a re ac t i on in p o s s ib l e f or arb i t rary value s

of the flux m , s at i s fying Talc m Tilcl

.

2

The re ac t i on p r oc e s s in a s t rong ( l im i t type ) de t onat i on

following the shock has i t in c ommon wi th a we ak de f lagrat i on that

unburnt gas in a s ub s on ic s t at e i s t rans forme d int o burnt gas in a

s ub s on i c s t ate . The two p r o c e s s e s di f fe r,howeve r , in othe r

re s pe c t s . The init i al t empe r ature in a de flagrat i on i s b e low

s afe ty tempe rature whi le the re ac t i on in a de t onat i on b egins wi th

a highe r t empe r ature . Al s o in a de flagrat i on the rate s of change

dT/dx ,d8/dx , d e/dx are z e r o ini t i al ly whi le the re i s no s uch

re s t r i c t i on for the re ac t i on pr oc e s s f ollowing a shock ; for small

value s of A and u the rat e s of change unde rgo s uch g re at change s

1Whe ther or not BE KE J/ae change s s ign during thi s pr oc e s s i s

immat e ri al,

c f . howeve r,v . Neumann 's re por t in whi ch trans i

t i on pr oc e s s are di s c us s e d on the b as i s o f e quat i ons ( 1 ) and ( 2 )(D

For nume ri c al de t e rminat i on of s uch p r oc e s s e s and va ri ous det ai le d d i s cus s i ons s e e the re p or t s by E yring and hi s c ol lab orat or s ,[ 5 ] ( whe re an exc e s s ive ly hi gh re ac t i on rat e wa s as sume d ) and [ 6 ]( The c as e shown in Fig . loa ,

p . 44 in [ 6 ] and labe le d " s te adyde f lagrat i on" the re is a we ak de t onat i on" in our te rminology . )

https://www.forgottenbooks.com/join

8 O and T O ev i dent ly re pre s ent the two s t at e s on b o th s i de s

of a p o s s ib l e shock t rans i t i on in the gas mixture charac te r i z e d by

the value s of e c ons i de re d . The re f ore , one of the s e two p oint s,

As’ c o r re s p ond s t o a s upe rs oni c , the othe r , B

8’ t o a sub s oni c flow .

We as s ume the value s o f the c ons t ant s P and Q s uch that the sur fac e

£1

int e r s e c t s the ini t i al p lane 2 O at two poin t s AO

and B0wi th

8 O, T 0 . From the d i s c us s i on on p . 3 3 i t foll ows that fo r

eve ry 2 0 two po in t s of int e r s e c t i on A and Be

exi s t as l ong ass

A or B d o not be c ome s oni c , or , what i s e quivalent , do not2 s

c oale s c e . We as sume that th i s i s not the c a s e f or O s 1 ; thi s

as sumpt i on i s pr imar i ly a c ondi t i on on the f lux m . We al s o as sume

that 8 O , T O at the s e p oint s f or O s 1 . We then have two

curve s of and f of po int s AS

and Be

along whi c h 8 and T are c on

t inuou s func t i ons of the parame te r s . ( The las t as s umpt i on made

he re is s omewhat s t ronge r than ne c e s s ary . For the di s cus s i on of

we ak de flagrat i ons , for examp le , only the exi s t enc e of the curve ; )

is nee de d . Inc i dent ally , the exi s t enc e of the supe r s oni c s t at e A8

wi th 8 O , T 0 alway s impl i e s the exi s t enc e of the sub s oni c

s t at e Be’ but the c onve rs e i s not t rue . )

A t the p oint s As’ B we have dT de 0 , d s/dx 0 exc ept

8

for s l or 8 88; henc e the fi e l d ve c t or p o in t s in the negat ive

e- d i re c t i on at the s e po int s . The p ro j e c t i ons of the fi e ld ve c t o rs

on the p lane s e c ons t . have s ingular i t i e s at the p oint s A8

and Be

' At a p oint A8

the p ro j e c t e d fie l d has a nodal point and

the s oluti on c urve s of the p ro j e c t e d fi e ld le ad f rom the ne ighb or

hood of thi s po int int o i t . At the p o int Be

the p ro j e c t e d fi e ld

has a s addle - s ingular i ty ; ( s e e Figure s 2 t o At the p oint s A6

96

Figure 2

De tonat ion

( integ ral curve s of the p ro j e c te d ve c t or fie ldon the plane 6 l fo r a de tonat i on )

Figure 3

De t onat i on

( Inte gral c urve s of the p ro j e c t e d ve c t or fie ld onthe p lane 6 c ons t . fo r a de t onat i on )

https://www.forgottenbooks.com/join

Figure 5

De flagrat i on

( Int e gral c urve s of the pr oj e c te d ve c t or f ie ld onthe p lane 2 c ons t . for a de flagrat i on )

2 - 4 1

and Be

on the s ur face 2 1 and in the re gi on 8 8S

the three

d imens i onal ve c t or fi e ld has s ingular i t i e s .

If we are inte re s t e d in a de flagrat i on we mus t as sume that

8 as

for the ini t i al pointBO , ( see p . i f we are inte re s t e d

in a de t onat i on we mus t as sume , ( s ee p . 33 and Figs . 2 , that

8 as

at the point Bo’ whi c h in thi s c as e is c onne c t e d wi th the

init i al p oint AO

thr ough a curve re p re s ent ing a shock . We re qui re

s omewhat more fo r de t onat i ons , viz . that 8 8S

on the whole

de tonat i ons are pos s ib le . Bo th p roc e s s e s have i t in c omm on that

the flow in the burnt gas i s s upe r s oni c . The s tat e ( 1 ) thus

b e longs t o the c as e ( A ) , and c or re s pond s t o a point A1

. As was

shown e ar li e r the re exi s t s in c as e ( Al )only a one - parame t ri c s e t

of s olut i ons whic h are regular at x 0 0 , and the value s of the

par ame te r may be chos en arb i t rari ly pr ovi de d i t is s o cho s en that

2 de c re as e s a s x de c re as e s . Thus the re exi s t s only one s olut i on

curve , C.

, s t art ing at Alwhi c h c ould rep re sent one of the

pr oc e s s e s men t i one d . If thi s s olut i on c urve (9 re ache s the point

B0

i t re pre sent s a s t rong de flagrat i on , i f i t re ache s the point Ao

i t re pre s ent s a we ak de tonat ion .

In the following we shall c ons ide r prob lem s di ffe ring in the

re ac t i on rate 8 ,all o the r parame te r s b e ing un al t e re d ; we may , for

example , as sume the fac t o r 800

( c f . 1 on p . 1 3 ) t o va ry from O

to a ) . The c urve s an d )3 are evi dent ly inde pendent of S ; the

curve C'

, howeve r , de pend s on 8 . If we want t o emphas i ze thi s

dependenc e we wr ite

2 - 4 2

I f the re ac t i on rat e S is ve ry high c ompare d wi th po/A and

pO/u, all ve c t or s po in t ne arly in negat ive s - d i re c t i on exc ep t ne ar

a l and 8 8S

In thi s c as e , the re fore , any s olut i on curve is

appr ox imate ly a s t rai ght l ine in the ne gat ive e- d i re c t i on unt i l i t

me e t s the surfac e 8 88

. The re fo re , the c urve C? that b eg ins at

the p o int Al’ i . e . at s 1 , 8 8

1 , T Tl , end s up on s O

ne ar ly wi th the value s 8 81’ T

Thus we se e that (1

end s up on e O at a p oint w i th 8 8S

i f S

Tl, henc e wi th ne ar ly 8

is suff i c i ent ly high . At such a p oint d e/dx O and henc e the re i s

n o c ont inuat i on of the c urve on the p lane 2 O and henc e none o f

the de s i re d ini t i al s t at e s i s r e ache d .

Le t u s c ons i de r the opp o s i t e ext reme that the re ac t i on rate

S i s ve ry small ; then the di re c t i on of the fie l d ve c t or l i e s eve ry

whe re ne arly in the p lane 2 c ons t . e xc ep t near the curve s L i

and fig. C ons i de r any " cy lindr i c al ne i ghb o rhoo d of the l ine u? and

exc lude from i t an arb i t rar i ly smal l ne ighb orhood o f the p oin t Al

‘

If S i s smal l enough then c le arly the ve c t o r f ie ld on the late ral

surfac e of the " cy linde r " p oint s int o i t s inte r i o r . The curve (3

,

b eginning at Al, c an the re fore neve r le ad far away f rom the c urve

ti fo r , as s oon as move d away fr om L.) the rate of change d e/dx

would be c ome much smalle r than and the re f ore the

curve 69 would again be drawn ne are r int o the ne ighb orhood of t ii.

C ons e quent ly,

the curve (9

mee t s the sur fac e 8 88not far f r om

the inte r s e c t i on AS

o f thi s surfac e wi th the curve c f, remains

from the re on the p lane 8 c ons t . and s oon ente r s the point As

'

In gene ral the c ours e of the curve C; c an be de limi te d as

f ollows : Le t b e the c oo rd inate s of the po in t A8

From the

1 0 2

https://www.forgottenbooks.com/join

p oint wi ll m ove c ont inuous ly along the te rminal l ine f rom the p lane

8 0 into the surfac e 8 88

. C ons e quent ly the re i s a par t i cular

value of S ofSOO f or whi c h the t e rm inal p o int l ie s on the int e r

s e c t i on of the p lane 6 0 and the s ur fac e 8 88

. We then s pe ak

of the " ex treme s i tuat i on and the ”ext reme " te rminal point T°

.

O the rwi s e , i f the t e rm inal p oint lie s on the surfac e 8 aswe

s pe ak of a " no rmal s i tuat i on , i f i t li e s on s O , of an

" abnormal" one .

In the ext reme s i tuat i on a we ak de t onat i on exi s t s ; for ,

s inc e 8 as

at the ext reme te rminal point , the s olut i on c urve

s t art ing at thi s po int remains on the s urfac e a 0 . That thi s

s olut i on curve end s up at AO

foll ows f rom the fac t that the fi e ld

has an at t rac t ive s ingulari ty at AO

and that the te rm inal p oin t

T.

l i e s in the c e ll T T'

T pl o

be l ong s t o tho s e po int s that are at t rac t e d by AO

. I t i s c lea r that

0p p 1

and henc e

a s olut i on curve c an re ach AO

only i f i t s te rminal po int li e s on

ext reme s i tuat i on , i . e . i f the re ac t i on rate S as sume s a part i c ular

high value S*

.

The d i s cus s i on of the te rminal l ine 7 wi ll al s o b e us e ful

for the inve s t igat i on of s t r ong de t onat i ons . Be fore we ente r the

d i s cus s i on of s t rong de t onat i ons and weak de flagrat i ons we mus t

inve s t igat e the po s s ib le c ont inuat i ons of s olut i on c urve s afte r

they have ente re d the re g i on 8 88

.

In the re gi on 8 88

the ac tual ve c to r fi e ld agre e s with the

p ro j e c t e d ve c t or fie ld . Supp o s e the po int Be

on the

curve 6) lie s b e l ow 8 8 i . e . suppo s e 88

as

Then the ve c t or8)

1 04

fi e l d ha s a s addle s ingul ar i ty at Bs’ ( see p . Gons e

quent ly , two s oluti on c urve s , £9: 2

Fig . and two s olut i on curve s , and ente r B

and £32, le ave B ( see

( thes0

5c urve s Wi th 8 88

are Q's

and t s ’ the one Wi th 8 88are L s

and We see from the ve c t or fi e ld ( Fig . 3 and Fig . 5 )

that if; end s up at the p o int A8

on. /Q .

A s olut i on curve 27 c an ente r the regi on 8 as

only at a

p oint where d8/dx O , hence only on the s e c t i on if o f the sur

fac e 8 88

cut out by the surfac e 5'

( s e e p . The inte r

s e c t i on of {f wi th a p lane 6: c ons t . wi l l b e denote d by 1

’

s,

( see Fig . 3 and Fig . We c ons i de r the c ont inuat i on o f the

s olut i on c urve df afte r it s ent ry int o 8 8S

on the s e gment

JZg The re are two c as e s . E i the r , the p oint Be

li e s ab ove the plane 8 8 i . e . 8 8 Then the c ont inuaS’ s 5

t i on o f 7 f rom { son , end s up in the point A

s’ a s s een from

the ve c t or fi e ld ( F ig . Or , the p oint Bel ie s in the regi on

8 88( see Fig . Then the curve

”7+

ente ring Bewi th de

e

c re as ing 8 inte r s e c t s the s egment ,Z; in a point Ge

' If . I

ente r s “Z5

on one s id e of Ge( on the s i de wi th large r T ) , i t s

c ont inuat i on end s up at As

a s be fo re . If A ) ente rs ~J€

on the

othe r s ide i t s c ont inuat i on end s up on T 0 un le s s i t has le f t

the regi on 8 8Sb e fore re ac hing T 0 . If .7 ente rs Z; at G

s

the c ont inuat i on of xi le ad s along ifs

t o Be

and from the re on

“[ c an be e i the r c ont inue d along L) ; up t o As

or along Z_; up to

T 0 unle s s j r

+ le ave s 8 asb e fo re re aching T O .

and Chapman- Jougue t de tonat ions imply sub s oni c or soni c flow in

the burnt ga s at x d ) . Hence c as e ( Bl) ob tains , o r the s t at e ( 1 )

c o rre sp ond s t o a po int Bl

C ons e quent ly , ac c or ding t o the s t at e

ment s made e arl ie r , ( see Se c t i on 3 , p . 2 1 ) the re is a two - parame t ri

s e t of s olut i ons re gular at x a ) . Henc e the re is a one — para

me t ri c s e t of s olut i on curve s Cf le aving the p o int B We want t ol

'

i n 0

show that among the curve s L the re I S a one ) c whi ch end s up at

AO, and thus re pre s ent s a s t rong ( o r Chapman— Jougue t ) de tonat i on .

The s e t of curve s 1: i s limi t e d by two curve s remaining on

the p lane 2 1 . One of the s e two c urve s , le ad s t o large r

value s of 8 , and end s up on the pl ane T O at a p o int Hl (

s e e

Fig . The othe r c urve , A’ l ’ le ad s t o smal le r value s of 8 and

e nd s up in the po int A i t re p re s ent s the po s s ib le sho ck t rans i1,

t i on from s t ate Al

t o s t at e B1

' The s e t o f c urve s may be

charac t e r i z e d by a parame te r 8 , whi c h for 8 0 y ie ld s ( J ; and for

8 1 y i e ld s X31 .

On the b as i s of the as sumpt i on made e ar li e r ( p . 4 1 ) that the

c urve (3 l ie s ab ove the p lane 8 8Swe s ee from the remarks made

b e fore that all c urve s A.that ent e r the p lane 8 8

Send up on u ( ;

none the re f or e me e t f1 b e l ow 8 8S

or end up on 8 0 . The re fore

the curve s At

end up e i the r on T O , s O , or on at . The s o

de fine d end point s w i l l b e c al le d ul t imate " point s and denote d by

U

B( S ) or U

S( S ) .

We inve s t igate whe the r or not the po int UB

de pe nd s c ont inu

ou s ly on the parame te r 8 . The c ont inui ty o f U6

c ould be inte r

rup t e d only in thre e c a s e s . The fi r s t c as e would be that UB

we re

the ul t imate po int of a curve (j whi c h pas s e s thr ough a s add le

s ingulari ty of the d i f fe rent ial e quat i on . Thi s po s s ib i l i ty is

https://www.forgottenbooks.com/join

c oinc i de s wi th the c urve (f. The ult imat e p oint U

1, the re f o re ,

l ie s on e O wi th 8 88

only if thi s is the c as e for the t e rminal

po int of Cf, viz . in the " abnormal " s i tuat i on . In the no rmal

s i tuat i on no te rminal p o int on s O wi th 8 8Sexi s t s . Cons e

quent ly , in the normal s i tuat i on the re exi s t s an ult imat e p oint U* ,

s O wi th 8 8S

The c urve £w= .tg whos e ul t imat e po int i s Up

remains on the

p lane 8 0 afte r having pas s e d thr ough U* , and ente r s the p oint

AO

. I t i s evi dent that the c urve i re pr e s ent s a s t r ong de t ona

de tonat i on i s p o s s ib l e . It i s not s o e as i ly s e en whe the r or not

s t rong de t onat i ons are p o s s ib le in the abno rmal c as e s inc e i t i s

not obvi ous whe the r or no t the re are ul t imat e p oint s on e O wi th

8 8S

f r om whi c h the c on t inuat i on lead s int o AO

.

Supp o s e the value of the re ac t i on r ate S i s s uch that we are

in a normal s i tuat i on but ne ar t o the ext reme s i tuat i on . Then the~X

ult imate p oint U i s ne ar t o the t e rm inal p oint Tfin of the curve C?

Henc e the c urve )C9

end ing up at U* wi ll fi r s t li e ne ar t o the

curve ( 71le ad ing fr om B t o A

land then ne ar t o the curve 4

1

le ading from Ai

t o T6

' Thus we s e e : in the ext reme c a s e , in whi ch

X

the p o int t fa ll s on T5 , the s t rong de t onat i on i s r epre s ente d by

r\

a c urve c ons i s t ing of t he c urve ( 31

then by' g

p

lea ding f rom Al

t o T6

and finally by a s e c t i on

in the p lane 6 0 f rom B1

t o

Al’

le ading fr om T6

t o AO

. In o the r words : in the ext reme s i tuat i on

normal but ne arly ext reme s i tuat i on the s t r ong de t onat i on wi ll b e

1 0 8

appr oximate ly a we ak de t onat i on fo ll owed by a s hock . The ext reme

s i tuat ion , howeve r , wi l l oc c ur only for ext reme ly high re ac t i on

rate s .

Le t us c on s ide r how a de t onat i on l ook s in the l imi t ing c as e

when the re ac t i on rate appr oache s z e r o ; a s far a s the curve s in

the - s pac e are c onc e rne d thi s i s e quivalent t o as suming that

A and u app r oach z e ro s inc e ASOO/p

Oand uS

OO/p

Oare the e s s ent i al

d imens i onle s s parame t e r s . The f ie ld ve c t or s in thi s c as e li e

almo s t in the p lane s 2 c on s t . exc ep t on the l ine s 2 4 and £3 . As

was s tat e d e ar li e r , the pro j e c t i on of the ve c t or f ie ld in the

p lane s s c ons t . ha s a s add le s ingulari ty at the po int s Be

' Henc e

the re are two l ine s A; and 13 ; de pending on s , whi c h le ad out of

Be

’ For 2 1 they c oinc ide wi th if

; and 13 1 . The curve Afe

le ad s t o l arge r value s of 8 , whi le I le ads t o smalle r value s

of 8 . In the lim it S ' O , or , in dimens i onle s s fo rm , uSG D/p

o

-fl - O

and ASOO/p

O

-fl - O , the s olut i on curve s 5 le aving the p oint Bl

c ons i s t of s e c t i ons of the curve .U foll owe d by the curve s f):or

i, ; unt i l the s e me e t the plane T O or the line From thi s

remark i t i s c le ar that the ult imate p oint s UB

UB( O ) on the plane

5 O c ons i s t of the curve on e 0 f rom the pointHO on the

plane T O t o the po intBO , then of the c urve up to the point

‘l'

AO

. I t i s c le ar from thi s de s c ri p t i on that the curve AV ( 0 ) con

s i s t s of the c urve i) from B1

t o B0

followe d by A 8 on e 0 up

t o AO

In the lim i t c as e us/pO

-» o, xs/po - o, the re fore , the

p roce s s , a s expe c t e d .

1

unle s s the re ac t i on rate exc e e d s a c e r t ain b ound . To thi s end we

s hould find out whe the r or not any of the curve s 49 s t art ing at

the point Bl

c an end up at the po int B0

. The s i tuat i on di ffe r s

f rom that for de t onat i ons in that i t mus t now b e as s ume d that

8 8Sat B

0’ ( s e e p . 4 1 and Fig . As a c ons e quenc e the line

7( ( S ) of ul t imate p oin t s U5( S ) i s m od i f i e d fo r small value s of S .

Fo r large value s of S , the s i tuat i on is as be fo re . F or S o) , the

ini t i al part of the ult imate l ine JTKS ) le ad s f rom t he p oint H1

on

T 0 s t raight ove r t o the p ro j e c t i on of H1

on s O . Fo r large

value s of S , the re f ore , the ul t imate line TI( S ) al s o b e g ins at H1

and le ads on the p lane T 0 ove r t o the p lane 8 0 . From the re

on U le ad s on s O t o the p o int3

16

i f 8 8S

at Al ’ or t o a

p oint U*

if a as

at Al

.

For small value s of S , howeve r , the s i tuat ion i s qui t e

d i ffe rent . We c an no longe r as s e r t that the ul t im at e line le ad s

ove r fr om Hl

on the p lane T O t o the p lane 2 0 . A s a mat te r

of fac t , fo r smal l value s of S the ul t imate line s t op s on T O at

l

a p oint with e O and j ump s d i s c ont inuous ly ove r t o the l ine Ji ;

the re as on b e ing that a curve,Q' exis t s whi ch ente r s the l ine

be l ow 8 8 ( thi s p os s ib i l i ty wa s exc lude d fo r de t onat i ons , ( s ee8;

PP We fi r s t c ons i de r the c as e S O . In that c as e the

c urve s A} c ons i s t of s e c t i ons o f the curve J5 fol lowe d by s e c t i ons

If thi s p i c ture of a de t onat i on is ac c e p t e d , then v . Neumann 'sre s ult tha t

fino we ak de tonat i ons exi s t i s imp li e d by the fac t that

the c urve 59 c onne c t s the po intBO wi th B

1and not w i th A

l'

https://www.forgottenbooks.com/join

The argument p re s ente d hold s j us t as we ll i f the po int s Al

and B c oale s c e s o that the fl ow c orre s ponding t o thi s p o int is

In the l im i t ing c as e whe re m and S OO app roach z e r o in s uch a

way that s S /me

app r oache s a fini te value , a de flagrat i on i s

always p os s ib le f or an appr op r i ate value of SG D/m2, as wa s shown

in Se c t i on 3 . In thi s l im it ing c as e d8 O and d s o f or the

ve c t o r f i e ld exc ep t on the p lane 8 PT , on whi c h the ve c t o r fie ld’\ J

is given by ( s e e p . The po int s B0

and B1li e on

I i s the l ine 8 81

on s 1 wi th

de c re as ing T . The l ine,72 i s the l ine 8 PT on e O wi th in

thi s p lane 8 pT . The line A,

c re as ing T . The point G0

i s then the inte r s e c t i on of 8 PT wi th

8 88

. As i s e as i ly s e en the re i s only one l ine on the plane

8 PT and thi s line mee t s the p o intGO only for a s p e c i al value

30

of s S/m2

.

Supp o s e we le t m inc re as e hold ing the init i al s t at e ( To, 8

o)

fixe d . Le t S (m ) b e the value of S fo r whi ch a de f lagrat i on exi s t s .

We have not pr ove d that S de pend s c ont inuous ly on m , but we c an be

sure that a c ont inuous c urve of p oint s (m, S ) in a (m, S ) — p lane

exi s t s re pre s ent ing pai r s of value s of m and S for whi ch de f lag ra

t i ons are p o s s ib le . E ventually the flux m wi ll re ach a value 111C

for whi ch the po int s A and B c oale s c e and a Chapman- Jougue t1 1

s i tuat i on ari s e s . Le t SCbe the c orre s ponding value of the re

1 1 2

APPE NDIX I

Of The Re ac t i on Pr oce s s

The c ons i de rat i ons of thi s report re s t on the b as i c

as sumpt i on ( see p . 3 ) that the re ac t i on pr oc e s s may be c ons i de red

app r oximate ly a sharp d i s c ont inui ty ; more s pe c i fi c ally,

that,

fi r s t ly , th e rate s of change of the pe rt inent quant i t ie s in the

fie ld of f low out s i de of the re ac t i on z one are negl igib ly smal l

when c ompare d t o the rate s of change of the s ame quant i t i e s ins i de

the re ac t i on z one and , s e c ond ly , that the rate of change of the

wi dth of the re ac t i on z one i s smal l when c ompare d wi th the ave rage

s pe e d wi th whi c h the gas e s c r o s s the re ac t i on z one . It was furthe r

as sume d ( p . 2 1 ) that the flow , when ob s e rve d from a frame mov ing

wi th the ins t ant ane ous ve loc i ty of the re ac t i on f ront , i s s t eady

in the ne ighb orhood of the re ac t i on fr ont at the t ime c ons i de re d .

We f i r s t want t o s how that thi s lat te r as sumpt i on i s c on

s i s tent wi th the as sumpt i on o f the d i s c ont inuous c harac te r of the

re ac t i on . Suppos e we de fine the ve loc i ty x of a b orde r of the re

ac t i on z one uti-xu

XO , and s upp os e we choo s e the ve loc i ty of our

f rame such that x O at one po int ins ide the re ac t i on z one . Then

the as s ump t i on that the rate of change of the wid th of the re

ac t i on z one i s sma ll c ompared wi th the ave rage ga s ve loc i ty in the

z one c an then be fo rmulate d as lxl u eve rywhe re in the zone .

C ons e quent ly , lu t l qxl eve rywhe re in the zone . The te rm u

t,

c an , the re fore , b e omi tt e d f rom the d i ffe rent i a l e quat ion whi c h

exp re s s e s the fac t that the ac c e le rat i on ut+ uu

xe qual s the tot al

1 1 3

app lie d for c e pe r uni t mas s . For s imi l ar re as ons one c an omi t the

t a m S T 8t ’ t ’ and a

tf r om the d i ffe rent i al e quat i ons exp re s s ing th

b alanc e o f mas s fl ow , ene rgy , and chem i c al re ac t i on . In o the r

word s,

t o the degre e of ac c uracy impl i e d by our b as i c as s ump t i on,

the e quat i ons c harac te r i z ing non- s t e ady flow r e duc e t o the e qua

t i ons in add i t i on t o m c ons t . Thus our a s sump

t i on of " loc al s t e ad ine s s is in agre ement wi th our ”b as i c

as s umpt i on .

S e c ond ly we want t o ment i on that fre quent ly re ac t i on flow

p r oc e s s e s oc c ur in whi c h our b as i c as sump t i on i s not s at i s fi e d .

De t onat i ons , c ons i s t ing o f a c hemi c al r eac t i on pr oc e s s ini t i at e d

by a shock , are f re quent ly fo ll owe d by rare fac t i on wave s . I t i s

c le ar that thi s rare fac t i on wave inte rfe re s wi th the re ac t i on

pr oc e s s , but the int e rfe renc e c an b e ignore d i f our b a s i c a s sump

t i on i s s at i s f i e d and the change s whi ch the rare fac t i on wave p ro

duc e s in a s e c t i on of the wi dth of the re ac t i on z one are signifi

c ant . If , howeve r , the re ac t i on z one i s s o wi de that thi s int e r

fe renc e c an no longe r b e igno re d , then p re s s ure and t empe rature in

the re ac t i on z one are d im ini she d and the s t rength of the init ia tin

s ho ck is re duc e d . In par t i c ular , the s pe e d of the de t onat i on wave

is then le s s than that c alc ulate d wi thout int e r fe renc e and c ould

thus b e le s s than that of a Chapman- Jougue t de t onat i on . If the

int e rac t i on is s t r ong the de t onat i on may eventually c e as e .

The oc c urrenc e of a l owe r de t onat i on l im i t may b e explaine d

in thi s way , s inc e the re ac t i on rate i s l ow , and hence the re a c t io

z one is wide , i f the c onc ent rat i on of the c ombus t ib l e c omp onent in

the unburnt exp lo s ive mixture i s l ow . ( Se e Wend land t

https://www.forgottenbooks.com/join

and not t o sho ck d i s c ont inui t i e s . No the ore t i c al t re atment of suc

p r oc e s s e s s e em s t o exi s t .

The s e remarks are inte nde d t o show that the fre quent ly

ob s e rve d devi at i ons fr om the p re d i c t i ons of the di s c ont inui ty

the ory are not due t o the uns t e ad ine s s of the p roc e s s a s such but

rathe r t o the oc cur renc e of a re lat ive ly wide re ac t i on z one whi c h

pe rm i t s t he int e rfe renc e of the out s i de fl ow wi th the re ac t i on

p r oc e s s .

APPE NDIX I I

It wa s shown in the text that fo r an exc e s s ive ly high

re ac t i on rate S S ( p , 8 ) a we ak de t onat i on oc cur s ins t e ad of the

Chapman- Jouguet de t onat i on ( s ee p . It i s of int e re s t t o know

how high a re ac t i on rate mus t b e in orde r t o b e exc e s s ive in thi

s ens e . The exc e s s ive c as e s are s eparate d fr om the re gular one s by

" maximal c a s e s in whi ch a Chapman- Jougue t de t onat ion i s j u s t

po s s ib le , whi le s uch a de t onat i on is imp o s s ib le for a re ac t i on rat

highe r than a maximal one . We shal l p re s ent s ome s uch maximal re

ac t i on rat e s nume ri c ally . From the s e re s ul t s i t wi ll appear that

fo r maximal re ac t i on rat e s the t rans i t i on z one b e c ome s ext reme ly

smal l , of the o rde r of magni tude of one mean fre e path , i f v i s c os

i ty and he at c onduc t i on are tho s e of ai r at 3 OOOK and atm o sphe ri c

p re s s ure . Unde r the s e c i rc um s t anc e s the not i ons of vi s c o s i ty and

he at c onduc t i on in the t rans i t i on z one be c ome meaning le s s . If ,

howeve r , vi s c o s i ty and he at c onduc t i on are ten t ime s as large as

for atmo s phe ri c ai r at 3 OOOK , exc e s s ive re ac t i on rat e s may we ll be

p o s s ib le . For examp le , S would the n be exc e s s ive i f i t vani she s

up t o 9 OOOK and e qual s s e c

- 1or m ore f or highe r tempe ra

ture s . The wid th of the t rans i t i on z one would then be ab out ten

t ime s as l ong as a me an free path .

We as sume that unburnt and burnt gas are polyt ropi c wi th

exponent s yoand For the mixture c ons i s t ing of the

f rac t i on 2 of burnt ga s and l - e of unburnt ga s we de fine y by

Then we have for the ene rgy pe r unit mas s of the mixture the

exp re s s i on

in whi ch the libe rate d ene rgy pe r uni t mas s F is as sume d t o b e

F 80

whi c h c or re s pond s t o a value 665 c a l/gm i f the ini t i al tempe rature

is Moeo/R

O3 OOOH and the mole c ular we ight of the unburnt gas i s

MO

2 9 .

Ab out the he at c onduc t ivi ty A and vi s c o s i ty u we have made

the as sumpt i on

whi c h wa s pr opos e d by B e cke r We shall in par t ic ular c ons i de r

as re fe renc e value for u the value

u 2 44 . 1 0‘ u

o gm/cm s e c

w i th y yo

the c orre s pond ing value of A i s then

A 8 -540

gm/cm s e c

We re c al l that the c us t omary c oe f fi c i ent s of vi s c o s i ty and he at

ac onduc t iv i ty are — ii , and RA in our no tat i on ; ( s e e f oo tnote 2 on

3

p .

The re ac t i on rate S was a s sume d t o b e z e r o up t o a s afe ty

tempe rature 8 85; fo r 8 8

8we have a s sume d

uS c ons t

thus , i f u is inde pendent of tempe rature and p re s s ure the s ame i s

then as sume d of the re ac t i on rat e f or 8 80

. Thi s as sumpt i on i s

rathe r unre al i s t i c be c aus e b oth the re ac t i on rate and the vi s c o s

i ty wi l l inc re a s e wi th the p re s s ure ; but thi s a s s ump t i on should be

suffi c i ent t o given info rmat i on ab out the orde r of magni tude of

maximal re ac t i on rate s . In Fig . 6 we have p lot t e d such maximal

re ac t i on rate s , or rathe r the value s £44 3 , in whi c h n

oi s the

re fe renc e vi s c o s i ty given ab ove . Fo r as afe ty temp e rature o f

9 OOOK or 8

S3 8 fo r examp le , we f ind as maximal value of the

ure ac t i on rat e S 1 6 -1 0 9

78 se c” 1

for 8 88

. The ve loc i ty V0wi th

whi c h the de t onat i on wave t rave l s int o the unb urnt gas at re s t ,

s o le ly de te rm ine d by t he Chapman- Jougue t c ond it i on , e qual s

v0

1 5°

7 m s e c

' lThe wi d th of the re ac t i on z one is r oughly given

by

son

I

I

<1

https://www.forgottenbooks.com/join

A maximal s i tuat i on , in whi ch the Chapman- Jougue t c ondi t i on

c an j us t b e s at i s f i e d , the re f ore c or re s pond s t o what was c alle d

an ext reme s i tuat i on , in whi ch the curve end s up on e 0

j us t wi th 8 88

. To ob tain a maximal re ac t i on rate we then pr o

c eed as foll ows . To a given ini t i al s t at e c or re s ponding

t o the p oint0’ we de te rm ine the end s t at e f rom the

Chapman- Jougue t c ond i t i on . The flux m i s then al s o de te rm ine d .

We now as sume any value for the re ac t i on rate S , or rathe r f or uS

and de te rm ine the curve The value of 8 wi th whi c h end s

upon 2 O i s then t aken as s afe ty t empe rature 88

. We finally

s e le c t th o s e value s f or the re a c t i on rat e S f or whi ch the s afe ty

tempe rature turns out t o b e b e twe en 6 OOOK and 9 OO

OK .

The curve ti wa s charac t e r i z e d a s the graph of that s olut i on

o f the di ffe rent i al e quat i on whi ch le ave s the p oint 3

1wi th de

c re as ing x . Al l o the r s o lut i on c urve s ente r ing T l ie on e l .

1

C ons e quent ly , u c an al s o b e charac te r i z e d a s the only s olut i on

c urve le av ing the p oint ‘ 11wi thout d e 0 . Henc e 2 c an b e int r o

duce d a s parame te r . The di f fe rent i al e quat ions then be c ome , afte r

int roduc ing t mgr

d t( l - e )uS d

—

Et ( p+ t — p l

- tl )

dp tt [ p t - l

‘

—1t2( y - 1 ) t t2 1

7 m 2 T: U

)

81

II

- 1 2

The de s i re d s olut i on t t ( e ) , p p ( e ) i s the n the one that

as sume s the value s ( tl ’ pl) fo r e l and pe rmi t s expans i on wi th

re s pe c t t o powe r s of ( 1 - s ) . I t is e as i ly obt aine d from thi s powe r

s e ri e s f or small value s of ( 1 - 6 ) and by fini t e d i ffe renc e s for

large r value s of 1 - s up t o e ==O .

Egi s is a re produc t i on of a NAVORD Report 79 - 46 , date d June

2 5 , 1 9

von Neumann , J . ,

" Progre s s Repor t on the the ory of de t onat i onwave s .

" John von Neumann c olle c te d work s , Vol . VI,

e di te d by A . H . Taub , Macmi llan Co . , New Y ork , N . Y .

C ourant , R . and Frie d r i chs , K . O ., Su e rs onic Flow and Shock

Wave s , Inte r s c i enc e , New York ( I948 ) .Be cke r , R . ,

"St os swe l l e und De t onat i on .

" Ze i t s chr ift furPhys i c , Vol . 8 , 1 9 2 2 .

Math . , 2 , 1 03

Parlin , R . B ., Duffy , G .

, Powe ll , R E . , and E yring , H . , TheThe o ry of E x losion Ini t i at i on . NDRC, Divi s i on 8 , OSRDNo . 2 0 2 6 , 1 9 3 . C onfident i al

E y ring , H . , Powe ll , R E . , Duffy , G . H ., and Parlin , R . B . ,

" TheChemi c al Re ac t i on in a De t onat i on Wave . NDRC, Divi s i on 8 ,

OSRD No . 3 796 , 1 944 . C onfiden t i al .

Lewi s , E . and von E lb e , G . , C ombus t i on , Flame s and E xplo s i on so f Gas e s . C amb r idge Unive r sity Pre s s , CamB ridge , 1 9 38

Jos t ,Wi lhe lm

, E xplo s i on s und Ve rb rennungs vorgAnge inGas en . E dwards B rothe r s , Afi fi Ar Bor , I9HS.

Semenov , N . N . ,

" The rmal The o ry of C ombus t i on and E xplo s i on .

"

NACA , Te c hni c al Memorandum No . 1 0 24 , 1 0 2 6 .

Boy s , S . E . and C orne r , J . ,

" The St ruc ture of the Re ac t i onZone in the Burni ng of a C oll oi dal Pro e l lant .

" Mini s t ryof Supply . A . C . ll3 9 , I . B . 8 , ( WA- 66 -49 Augus t , 1 94 1 .

Se c re t .

C o rne r , J . ,

" The ori e s of Flame - Spe e d s in Gas e s . ArmamentRe s e arch Depar tmen t . Oc tobe r 1 943 . The ore t i c al Re searchRe po rt No . ( WA- l297 Se c re t .

Wendland t , Rud olph ,

" Die De tona tionsg renz e in E xplos ivenGasgemi s che n . Z . Phy s . Chem .

, 1 92 5 , p . 2 27 .

Tay lo r , G . I . ,

" De tonat i on Wave s . Mini s t ry of Supply

.

E xpl os ive s Re s . C omm . , R . C . 1 7 8 , A . C . 63 9 ( W- l2 - l

Feb ruary 1 94 1 . Confident i al .

Chem i c al Kine t i c s

Pe te r D . Lax

C ourant Ins t i tut e of Mathemat i c al Sc ienc e sNew York Unive r s i ty

The p r opagat i on of chem i c al re ac t i on s in c ombus t i on is

gove rne d by the rate at whi c h ene rgy i s t ransp or te d in and out of

the re ac t i on z one , and on the rat e at whi ch the chem i c al re ac ti ons

pr oc e e d . In the s imp le s t m ode l s the re ac t i on i s as sume d t o pro

c e ed at an exponent i al rate exp [ kt ] , k be ing a fun c t i on of tempe ra

ture , changing from O t o s om e high value b eyond the s o— c alle d

igni t i on tempe rature . In re ali ty c hem i c al re ac t i on s are m ore

c omplex , as t oni shingly c omp lex ; a good un de r s t anding o f them i s

ne c e s s ary t o gauge the limi t at i on s o f S imp le mode l s and t o deve l op

more re al i s t i c one s . The purpo s e of thi s le c ture i s t o p re s ent

the e lement s of chemi c al kine t i c s . For a more thorough t re atment

we re c ommend a text on Phy s i c al Chemi s t ry s uch as f or the

s t at e of the ar t the Sympo s ium Proc e ed ing s c ont aine d in [ 6 ] s hould

b e c ons ulte d .

A chem i c al re ac t i on i s the f ormat i on o f one or s eve ral c om

pound s , c alle d pr oduc t s of the re ac t i on , out of one or s eve ral

c ompound s or e lement s c alle d re ac t ant s . Hous e hold examp le s are

2 H2+- 0

2-' 2 H

20

H2+ I

22 HI

Gene rally,denot ing re ac t an t s by M

jand pr oduc t s by N

j’

https://www.forgottenbooks.com/join

int e rme d i ate p roduc t s - at om s , fre e rad i c al , ac t ivate d s t at e s . Thi s

netwo rk of e lement ary re ac t i on s i s c alle d the re ac t i on me chani sm .

In orde r t o analyz e a chemi c al re ac t i on , one ha s t o pe rform

thre e t ask s

a ) Find all re levant re ac t i on me chani sm s .

b ) De te rmine the rate s at whi ch the e lement ary re ac t i ons

ent e ring a me chani sm p ro c e e d .

c ) De te rm ine the ove rall rat e at whi c h t he re ac t i on

m echani sm p roc e e d s .

Typ i c al re ac t i on me chani sm s may involve upward of 8 0 s pe c i e s ;

find ing all the re levant one s i s an art . We c all the re ade r s

at t ent i on t o the on- going c ont rove r sy ab out the rate at whi ch

f luor oc arb on s re le as e d by s p ray c ans are remove d fr om the uppe r

atmo sphe re ; the c ont rove r sy i s ab out po s s ib le re ac t i on me chani sm s

involving o z one and fluo ro c arb ons . The mos t int rigu ing as pe c t of

re ac t i on me chani sm s i s c at aly s i s , whe re the p re s enc e of a small

amount o f c at aly s t make s p o s s ib le a re ac t i on me chani sm whi c h pro

c eeds ext reme ly fas t,

at the end of whi c h the c at aly s t is re s t ore d .

We remark that s inc e e lement ary re ac t i on rate s are rap i d ly vary ing

func t i ons of t empe rature , a re ac t i on me chani sm that i s the re levant

one , i . e . the fas t e s t , at one t empe rature may b e i r re levan t at a

highe r t empe rature .

The de te rminat i on of rate s of e lement ary re ac t i on s is a

c ol lab orat ive e ff o rt b e tween expe rimente r s and the or i s t s . We s hall

s ay a few word s ab out the the ory , a c omb inat i on of s t at i s t i c al

me chani c s and quantum chemi s t ry . AS remarke d e ar li e r , an e lemen

tary re ac t i on c an take plac e only i f ene rgy , exc e e d ing the

1 24

ac t ivat i on ene rgy , i s s upp li e d . The s our ce of that ene rgy i s the

t rans lat i onal ene rgy of s uff i c i ent ly ene rge t i c m ole cule s ; upon

c ol li s i on trans lat i onal ene rgy i s c onve rt e d int o int e rnal ene rgy .

A s suming that part i c le s are s t at i s t i c ally inde pendent of e ac h

othe r , the fami li ar Stos s z a hl Ans at z , the numb e r of c olli s i on s wi ll

b e pr oport i onal t o the pr oduc t of the c onc ent rat i on of re ac t i ons .

Suppos e the re ac t i on is given by e quat i on le t 's denote the

c onc ent rati on of s pe c i e s Mjby [M

j] ’ of N

jby [ N

j] ’ me asure d in

mole s/cm}

. Then the rate at whi c h the s e c onc ent rat i ons change i s

Thi s i s c alle d the law of mas s ac t i on , and kf

i s c alle d the fo rward

re ac t i on rate . Many , the ore t i c al ly all , re ac t i ons go b oth fo rward

and b ackward ; the forward re ac t i on rate is denote d by kf

, the b ack

ward rate a s kb

, and the re ac t i on ( 1 ) is writ ten as

kf

5 Ni v“

E qui l ib rium i s e s tab li she d at such c onc ent rat i ons whe re the fo rward

and b ackward re ac t i on rate s are e qual . S inc e the law of mas s

ac t i on for the backward re ac t i on is

dM m

V

J

dV

JHf [ N

J] — v

jkb T_T [ N

J]

a t e qui l ib rium

u .

k T [M3] J = k

_[ [ N . ] J

J

The s e re lat i ons le ad t o an e asy de t e rm inat i on o f the rat i o kf/kb,

and only one of the two rate c on s t ant s kf

or kb

ne e d t o b e

me as ure d .

As we shall show be l ow , the re ac t i on rate i s an exponent i al ly

de c re a s ing func t i on o f the ac t ivat i on ene rgy . S inc e the ac t ivat i on

e ne rgy for the b ackward re ac t i on e qual s the ac t ivat i on ene rgy o f

the forward re ac t i on p lus the ene rgy re le as e d in the forward re

ac t i on , it foll ows that for re ac t i on in whi ch a g re at de al o f

ene rgy i s re le as e d , the b ackward re ac t i on i s negl ig ib ly s low .

S ince the numbe r of ene rge t i c par t i c le s who s e c oll i s i on le ad s

t o p o s s ib le chem i c al re ac t i on i s a rap i d ly inc re as ing func t i on of

tempe rature , s o i s the re ac t i on rat e . Ar rhenius ' law s t ate s that

k B e- E /RT

whe re E i s the ac t ivat i on ene rgy , R the gas c ons t ant and B a

c ons t ant . A mo re e lab orate s tat i s t i c al c olli s i on the o ry , taking

inte rnal deg re e s o f fre e dom int o ac c ount , give s

k

whe re B i s a func t i on of tempe rature , typ i c ally a p owe r of T . Rate s

c alculat ed thi s way are much highe r than expe riment al ly ob s e rve d

value s . The re as on i s that not all c oll i s i on s le ad t o a re ac t i on ,

only tho se whe re the c ol li ding mole cule s are p rope r ly or i ent ed .

Thi s c an be c orre c te d empi r i c ally by c ut t ing down k by a fudge

fac t o r c alle d a s t e r i c fac t or . A more s at i s fac t ory c alc ulat i on c an

https://www.forgottenbooks.com/join

k large po s i t ive , whos e exac t s olut i on i s e The c rude fo rward

s cheme

who s e s olut i on i s

le ad s t o

( 3 ) x ( n6 ) ( 1 ks )m

When k i s large , s ay C( loa) , the s olut i on ( 3 ) is exp onent i ally

un s t ab le , unle s s 6 d( lo-a) , a pr ohib i t ive ly small t ime s t ep .

A reme dy i s t o us e s cheme

who s e s o lut i on i s

lX<t+ 6 ) l i— k6

so that

l nx ( ns ) ( l i- k6 )

Thi s is s t ab le and app roximat e s we ll e

_kt, t ==n6 , regard le s s of

the s i z e of k .

We now t ry the impl i c i t s e c ond or de r s cheme

who s e s olut i on is

s o that

Thi s is s tab le , i . e . uni formly b ounde d , for all k and 6 ,but for

ktk large is not a go od approximat i on t o e

'

t ==n6 . What i s t rue

for thi s s imp le examp le is t rue for sy s t em s , and shows that t o

ob t ain ac c urate s olut i ons of St i ff sy s tem s one mus t us e a s pe c i ally

de s igne d nume ri c al s cheme . The mo s t ve rs at i le and b e s t known

me thod is due t o W . Ge ar , s e e Ge ar 's me thod is avai lab le as

a us e r- ori ente d package d pr ogram , se e

At the end of thi s t alk we wi ll show how t o exploi t s t i f fne s s

by mak ing u se of asympt ot i c me thod s .

We now give s ome example s

( 1 ) Re ac t i on : He

-r IQ

-fl - 2 HI

Re ac t i on me chani sm

Rate c ons t ant s

tuning ]

Rab

-

I ] [ H2 ]‘ k3[ H ] [ 1 2 ]

2kh [ 1 ]

2

g ram ?) karma )

( I I ) Re ac t i on : A -fi - B 4- C

Reac t i on me chani sm

Rate c ons t ant s

-X

( 4 ) A + A .- A 4 A kf, b

( 5 ) A*

- > B + C

A*

is a s o- c al le d ac t ivate d mo le c ule , forme d by c ol li s i on of

ene rget i c m ole c ule s A ; the ac t ivate d mole c ule A* de c ay s s p on tane

ou s ly int o the fragmen t s B and C .

[A ]— kf tAl + k [ A ]

( I II ) Re ac t i on : 2 03

-3 O2

Reac t i on me chani sm

( 8 )

( 9 ) o + 03

— + 2 o2

kft031 + k

btogit0 1 4 1 0 1 10

31

kf togi 2 s [ ON O ]

https://www.forgottenbooks.com/join

3 - 1 1

Sub s t i tut ing thi s int o ( 1 4 ) we ob t ain an e quat i on of s imple

exponent i al de c ay fo r m , g iv ing

( 1 7 ) m m c ons t exp

Us ing ( 1 5 ) and ( 1 6 ) we de duc e fr om ( 1 7 ) that

- skf

[ A ] m c ons t exp

We now turn t o c as e he re t o o k and kf b are ve ry much

large r than 5 ; he re t oo thi s fac t c an be exp lo i t e d t o g ive an

asymp t o t i c analy s i s of the las t s t age of the re ac t i on . As b e fore ,

the thi rd te rm in all thre e e quat i ons ( lo ) i s smal l

c ompare d t o the fi r s t and s e c ond te rm s , kf [ 03

] and

the re fore the e ffe c t of thi s thi rd t e rm i s neg ligib le unt i l the

fi r s t two t e rm s b e c ome ve ry ne ar ly e qual , i . e . unt i l we c ome ne ar

e qui lib rium for re ac t i on ( 8 )

We al s o as sume that oz one and oxygen at om c onc ent rat i ons are small

c ompar e d t o that of oxygen mo le c ule s . If s o , we c an c alc ulate the

value of [ 0 2 ]s inc e i s a c ons t ant in t ime .

Denote thi s value of [ 0 2 ]by Y ; from

( 1 9 ) [ 03] m KY [ O ] K k

b/kf

T o c alculat e the t ime hi s t or i e s of [ 03] and [ 0 ] add e quat i ons

and we ge t

( 2 0 ) m

whe re

( 2 1 ) m

Us ing ( 1 9 ) and ( 2 1 ) we ge t

[ O31 2 I

—

FKY ’

Sub s t i tut ing t hi s int o ( 2 0 ) give s

who s e s olut i on is

( l +- KY )

We now turn t o an asympt ot i c me thod deve lope d by c hemi s t s ,

s i tuat i ons whe re one of the c omp onent s i s ve ry ne arly in s t e ady

s tate , i . e . i t s c onc ent rat i on hardly c hange s wi th t ime . Thi s i s

the c a s e e . g . in ( I I ) , whe re the value of [A*

] is de te rmine d mainly

by the e qui lib rium o f the re ac t i on s in although A’ de c ays

s pon tane ous ly , the rate of de c ay s i s as sume d t o be small , s o that

e qui lib rium i s re e s t ab li s he d . The me thod c ons i s t s in as suming that

exac t s t e ady s t ate has b e en re ache d , i . e . that the t ime de rivative

of the c omponen t in que s t i on i s z e ro . The re s ult ing algeb rai c

re lat i on i s us e d to e lim inate one of the c onc ent rat i ons from the

1 3 3

sy s tem of d i ffe rent i al e quat i ons ; the remaining dim ini s he d sy s t em

o f ODE's 1 8 fre e of s t i ffne s s .

We now app ly thi s method t o sy s t em s e t t ing g%in ( 7 ) give s

2( 23 ) af ter S [A

*

] o

from whi ch

at

kf [A ]2

kf

i . e . A and A*

are ve ry ne ar ly in e qui l ib rium . Now add ( 23 ) t o

we ge t u s ing ( 24 )

who s e s olut i on i s

( 2 5 ) [ A ] c ons t exp

Thi s re s ult agree s wi th when k i s ve ry much large r than kh f ’

but d i s ag re e s othe rwi s e . Whi ch is c or re c t ? nume ri c al integ rat i on

o f the sy s tem ( 7 ) give s the nod t o thi s is not s ur

pri s ing s inc e one c an p rove rigorous ly that is t rue . In

the i r inte re s t ing art i c le " The S te ady S tate App roximat i on

Fac t or Farr ow and E de l s on analy s e a re ac t i on me chani sm

involving 8 1 e lement ary re ac t i ons ; the ODE sy s t em de s c r ib ing the

reac t i on i s s t i ff . They s olve thi s sy s tem by Gear ‘ s me tho d ; thi s

s o lut i on d i ffe r s s igni fi c ant ly fr om previ ous ly ob t aine d s olut i ons

us ing the s t e ady s t at e app roximat i on . S ince Ge ar ‘ s me thod is re li

ab le , thi s shows that the s t e ady s tat e me thod is not . Neve r the le s s

1 3 4

https://www.forgottenbooks.com/join

Far row,

L A . and E de l s on , D .,

" The S te ady S t ate App roximat i onFac t o r Int . J . of Chemi c al Kine t i c s , Vo l . 6 ,

1 974 ,

pp . 7 8 7 — 8 0 0 .

Ge ar , C . W . , Nume ri c al Ini t i al Value Pr ob lem s in Ord inaryDi ffe rent i al E quat i ons " , Prent i c e Hall , E nglewood C l i f f s

, N . J .,

1 97 1 .

Hindmar s h , A . C GE AR : Ord inary Di ffe rent i al E quat i on Sy s temS olve r" , Te chni c al Rep or t No . UOID 3 0 0 0 1 , Rev . 2 , LawrenceLive rm ore Lab o rat o ry , 1 97 2 .

Kre i s s , H Prob lem s wi th di ffe rent t ime s c ale s f or ord inarydi ffe rent i al e quat i ons , Re po r t No . 6 8 , Upp s ala Univ .

, Dep t .

of C omput e r S c i enc e , 1 977 .

Moore , W . J . , Phy s i c al Chemi s t ry , 4th

e d . , Prent i c e Hall ,

E nglewood C l i f fs , N . J . , 1 97 2 .

Symp os ium on Re ac t i on Me chani sm s , Mode l s , and C omput e r s , J .

Phy s i c al Chemi s t ry , Vol . 8 1 , No . 2 5 , 1 977 , pp . 23 09 - 2 5 8 6 .

C ombus ti on , Flame s and E xpl o s i ons of Ga s e s,

Lewi s , Be rnard andvon E lbe , Guenthe r

,Ac adem i c Pre s s ,

1 3 6

RANDOM C HOICE ME THODS WITH APPLICATIONS TO RE ACTING GAS FLOW

Alexand re Joe l Cho rin*

Department of Mathemati c s and Lawrence Be rke ley Lab orat oryUnive rs i ty of C ali forni a

Be rke ley , C ali fo rnia 947 2 0

Ab s t rac tThe rand om choi c e me thod is analyz e d ; appr op ri ate b oundary

c ond i t i ons are de s c ribe d , and app l i c at i ons t o re ac t ing ga s fl ow inone dimens ion are c arr ie d out . The s e app l i c at i ons i llus t rat e theadvant age s of the me thod .

Int r oduc t i on

The random choi c e me thod for s olving hype rb oli c sy s tems wa s

int roduc e d as a nume ri c al t oo l in It grew from a c ons t rue

t ive exi s t enc e p roof due t o Glimm In thi s me thod , the s olu

t i on of the e quat i ons i s c ons t ruc t e d a s a supe rp os i t ion of loc al ly

exac t e lement ary s imi lari ty s olut i ons ; the s upe rpo s i t i on i s c arr ie d

out through a s ampling pr oc e dure . The c omputing e ffo rt pe r me sh

point i s re lat ive ly large , but the glob al e ff i c i ency i s high when

the s olut i ons s ought c ontain c omponent s of wid e ly d i ffe ring t ime

s c ale s . Thi s e ffi c i ency i s due t o the fac t that the app rop ri ate

inte rac t i ons c an be p rope rly taken int o ac c ount when the e lement ary

s imi lari ty s olut i ons are c ompute d . The aim o f the p re sent le c tu re

i s t o pr ovide a furthe r analys i s of the me thod , and t o i llus t rate

i t s us e fulne s s in the analys i s of re ac t ing gas flow . E xample s are

g iven o f de t onat i on and de flag rat i on wave s , wi th infini te and

fini te re ac t i on rate s .

1 3 7

We beg in by de s c r ib ing the me thod b ri e f ly . C ons ide r the

hype rb oli c sy s t em o f e quat i ons

( 1 ) Vt

V ( x , 0 ) siven

when v i s the s o lut i on ve c t or , and s ub s c r ip t s denote di f fe rent i a

t i on . The t ime t i s d iv ide d int o inte rvals of length k . Le t h b e

a s pat i al inc rement . The s olut i on is t o b e evaluate d at the po int s

( ih , nk ) and i n Le t2

n n+ 1 /2 l 1

B iapp r oximate and app rox imate

n+ 1 /2 n nThe alg or i thm i s de fine d if c an b e found when u

i, E i+ l

are

known . C ons i de r the foll owing Riemann p rob lem

t O — d ) x + m>

Le t w ( x , t ) denot e the s o lut i on of thi s p rob lem . Le t 8 b e a. value

of a random vari ab le 8 , é . Le t Pib e the point ( e

ihig) ,

and le t

W w ( P w ( 8 h 1E )i i ’2

b e the value o f the s olut i on w o f the Ri emann p rob lem at

se t

https://www.forgottenbooks.com/join

i i ) n is as s ume d known in advanc e ; the int e rval é ,3 ] i s

d iv ide d int o n s ub int e rval s of e qual lengths and 81

i s p i cke d in

the mi d dle of the ith

s ub inte rval ;

i i i ) ( A c omp r omi s e b e twe en i ) and i s d ivi de d

int o m s ub inte rval s , m n , and 81

i s p i cke d at rand om in the

fi r s t s ub inte rval , 82

in the s e c ond s ub inte rval , 8m+ l

in the fi r s t

s ub inte rval , e t c .

A fourth s t rategy whi c h re l i e s on the we ll - e quipa r tit ioned

s e quenc e s s tud i e d by Rich tmye r and Os t r owski wa s s ugge s t e d by Lax

but is not us e ful in the p re s ent c ontext .

I f s t rategy i ) i s us e d , we have

x+ n d i s p lac ement of the ini t i al value

2n

whe re

9 i f he - k/22 i

- 2 i f h8i

— k/2

vari anc e ”i i s re ad i ly evaluate d

h2

K kva r ( Iii )

"

21( 1 '

H fll 'i‘

fi )

vari anc e n is thus

T ( l - fi fl u g)

and the s t andard devi at i on o f n, whi ch me asure s it s magni tude is

t<1 o (ffi h ) .

If the s e c ond s t rat egy is us e d ,

In!

i f n O ( h— l) , n o( h ) . If the thi rd s t rat egy is us e d , and n is

a mult iple of m , n s inc e only in eve ry mth

hal f s te p

is the out c ome of the s ampling in doub t .

As s ume v is of c ompac t support . Foll owing a s ugge s t i on by

Lax,we de fine the re s olut i on o f the s cheme by

nQl

min uiq

whe re deno te s the maximum no rm . The s cheme ha s re s olut i on o f

o rde r m i f Q O ( h-m) . The di s plac ement d of the s cheme i s de f ine d

by

- l nQ u

i

min | up

iq

The me thod app li e d t o the pr e s ent p rob lem ha s alm os t fi r s t orde r

ac curacy , almo s t f i r s t orde r d i s plac ement , but infini t e re s olut i on .

The re i s no smoothing and no nume ri c al d i ffus i on or di s pe r s ion .

Fo r any k/h ,the d omain of dependenc e of a po int i s alway s a s ing le

po int . The answe r s are always b ounde d . I f the C ourant c ond it i on

k/h l i s v i olate d , the e quat ion be ing app roximate d i s

vt

C le arly,

s inc e the se re s ult s a re independent o f k/h ,

they gene rali z e to hype rb ol i c sy s tems wi th c ons tant c oe ffi c ient s .

Cons i de r now the e quat i on

a ( x , t )vx

in - d l x + 6 3 , t O , v ( x , 0 ) g ( x ) g iven , and a ( x ,t ) a

Lip s chi t z c ont inuous fun c t i on o f b o th x and t . The me thod i s not

we ll s ui t e d t o the s olut i on o f such an e quat i on , b oth be c aus e the

s olut i on o f the Riemann p rob lem re qui re s a p o s s ib ly lab o ri ous int e

gra t ion o f a charac t e ri s t i c e quat i on , and b e c aus e the e rr or s wi ll

turn out t o b e large c ompare d with thos e incur re d in othe r avai lab le

me thod s . The analy s i s i s neve r the le s s i lluminat ing .

Le t CX

b e the charac t e r i s t i c0

95 x ( 0 ) xd t 0

F or e ach i , we have

i f P l i e s t o the right of C

i f P li e s t o the le ft of C

A s b e fore

x ih t nk

whe re n is a rand om variab l e whi ch now de pend s on b ot h x and t .

If 8 i s p i cke d at rand om from the uni form d i s t ributi on on

( S t rate gy i ) ) we have as b e fo re n S t rategy i i )

c le ar ly yie ld s an e r ror O ( l ) . S t rat egy i i i ) i s more advantage ous ;

the s t andard deviat i on o f n i s again b ounde d by Howeve r ,

the me an of n is no longe r z e r o . As s ume k O ( h ) . Note that

a ( x , t ) may vary by O (mh ) be fore thi s c hange affe c t s the value s of R

Thus , n me an of n O (mh ) , and n O (mn) i I f

1 4 2

https://www.forgottenbooks.com/join

and s imi lar ly fo rn

. x2- xl

k, 5 )

fe l l be twe en the two charac t e r i s t i c s . As sume the fi r s t

s amp ling s t rate gy i s us e d . The re are two s ourc e s of e r ror whi ch

has inc re as e d by ih e ach t ime

P = ( 8 h

make An 0 . The re i s the s t andard devi at i on of the s um of the

random variab le s whi c h e qual ih when P i s b e twe en the cha ra c t eris

t i c s , and are z e ro othe rwi s e , ( thi s is c le arly and the re

is the unc e rt ainty in the s l ope of the charac t e r i s t i c s due t o the

late ral d i s p lac ement of the s olut ion ; thi s i s again O (h/h ) and

induc e s an e rror i f n O ( h- l) , thi s is

Thus An and the re s olut i on i s not of highe r

orde r than the ac c uracy . S im i lar re sult s hold fo r the othe r

s ampling s t rategie s .

We now turn t o the nonl ine ar pr ob lem

vt

whe re f i s a func t i on of v but not exp li c i t ly a func t i on of x

and t . The me thod of analys i s we have us e d he re i s not app li c ab le ,

s inc e value s of v are not me re ly pr opagat e d along charac t e ri s t i c s .

Furthe rm or e , we have he re no w ay of t aking int o ac c ount p rope r ly

the fac t that rare fac t i on or l o s s of informat i on incurre d in the

nume r i c al pr oc e s s c or re s pond t o genuine p rope rt i e s of the di ffe ren

t i al e quat i ons . Al l we c an p r ovi de he re is a he ur i s t i c analys i s .

C ons i de r the thi rd s amp ling s t rat egy . S inc e the s lope of the

charac t e r i s t i c de pend s on the value s of v and not on x , the value s

of v at neighb orhing p oint s remain att ache d t o ne ighb o ring point s ,

we expe c t the t e rm O (mh ) in n t o d i s appe ar , and have n

1 44

Thus , the re s olut i on should b e at le as t Note that i f

” 1) and m O ( h ) , the random e lement in the me thod los e s it s

s igni fi c anc e .

In the c as e of a shock s ep arat ing two c ons t ant s t at e s,

one

c an re adi ly s e e that d 0 (h/h7m) but the re s olut i on i s infinit e .

One c an t riv ially de fine re s o lut i on in a ne ighb orhood . Thus , what

we have i s a rathe r awkward fi r s t orde r me thod , whi c h re s olve s

shock s ve ry sharply . We als o know that i t ke ep s fluid inte rface s

pe rfe c t ly s harp It i s us e ful for the analys i s of prob lem s in

c art e s ian c oo rd inat e s in whi ch the dynami c s of the di s c ont inui t i e s

are of p aramount s igni fi c anc e . We shall p r ovi de example s of such

p r ob lem s in late r s e c t i ons . Re c ent re s ult s ( see , e . g . , show

that in such pr ob lem s s ub s t ant i ally highe r ac curacy c annot be

achi eve d .

The c or re c t impo s i t i on of b oundary c ondi t i ons in our me thod

re qui re s c are ful thought,

and was not ade quat e ly di s cus s e d in

I t i s c lear that even in the c as e of e quat i on ( 2 ) the p re senc e of

a boundary c an de t rac t f rom b oth ac curacy and re s olution . The

late ral d i s plac ement of the s oluti on may make s ome func t i on value s

d i s appe ar ac ro s s the b ounda ry and c are mus t b e t aken t o ensure the

po s s ib i l i ty of the i r re t ri eval . Add i t i onal s t o rage ac ros s the

b oundary and c are ful ac c ount ing of the lat e ral di splac ement p rovide

a reme dy .

The fol lowing proc e dure has be en int roduced in [ 2 ] to reduce

the late ral d i s plac ement of the solut ion ( and thus reduce the los s

1 45

of info rmat i on at wall s ) , when the thi rd s amp ling s t rate gy i s us e d .

The goal is t o ob t ain as fas t a s p o s s ib le s olut i on value s on b oth

s i de s of what eve r wave patt e rn eme rge s in the s olut i on o f the

Riemann p r ob lem , and thus rap i d ly o ffs e t a d i s p lac ement t o right

by a di s p lac ement t o the le f t ( or vi c e ve r s a ) . We p i ck an intege r

m' m , and m and m' mutually p rime , and n inte ge r , n m,

andO 0

c ons t ruc t the s e quenc e of int e ge rs

( 3 ) m )

The sub inte rval s of are then s amp le d in the orde r

n n n rathe r than in the natural s uc c e s s i on . One c an0’ l ,

fur the r m od i fy the s amp l ing s o that of two s uc c e s s ive value s of 8 ,

one li e s in and one in The s e p roc e dure s d o not

inc re as e the e r ror far f r om the wall , and are qui te e ffe c t ive ,

al though no analyt i c al as s e s sment of the i r e ffi c i ency is avai lab le .

Supp o s e we are s olving the e quat i ons of gas dynami c s ( e qua

t i ons ( 4 ) b e l ow ) , and us ing the thi rd s amp ling s t rat e gy , mod i fi e d

by ( 3 ) or not . As s ume the ve loc i ty v i s given at the b oundary .

One c an f ind a s t at e ( i . e a s e t of value s fo r the gas vari ab le s )

whi ch has the given ve loc i ty and whi ch c an b e c onne c t e d t o the

s t ate one me sh point int o the f lui d by a s imp le wave ( s e e , e . g . ,

Thi s is e quivalent t o s olving half a Riemann p rob lem , and

p rovide s an app r opr i at e s olut i on fi e l d whi ch c an be s amp le d . The

s ame re sult c an be ob t ained by symme t ry c ons i de rat i ons . C ons i de r

a b oundary p oint t o the right on the re gi on of flow ; le t the

b oundary c ondi t i ons b e impo s e d at a p oint i h . A fake right s t at e0

at i s c re ate d,wi th

https://www.forgottenbooks.com/join

lo s t t o the wall , but the vari anc e of the s olut i on wi ll b e

inc re as e d . Be t t e r s t rate gie s c an be devi s e d , but re qui re thought

in each s p e c i al c as e . I f the wall s are at re s t , V 0 , one c an

p ro c e e d a s fo ll ows : imp o s e the b oundary c ondi t i on on the right at

h , and on the le ft at t ime ( n + l )k at a p ointt ime nk and a point i2l

l( 1 2 l

l, 1

2int ege r s . One c an s e e that i f 8

1, 8

2are s o

c ho s en that 81

O at t ime nk , and 82

O at t ime then 81

and 8 c an be us e d at the b oundary a s we ll a s in int e r i o r wi thout2

l o s s of re s olut i on .

Our g oal in thi s s e c t i on i s t o p re s ent a qui ck summary of the

e lementary the ory of one d imens i onal de t onat i on and de flagrat i on

wave s , ( fo r more d e tai l , s e e , e . g . , [4 ] and [ lo] , and then de rive

s ome re lat i ons b e tween the hyd rodynami c al variab l e s on the two

s i de s of suc h wave s for lat e r us e .

The e quat i ons of gas dynami c s are

( 4 a ) pti- ( pV )X

O

(ab ) ( pv ) t+ ( pV

2+ p )

X0 ,

( 4 c ) et-l O

whe re the s ub s c r ip t s denote di ffe rent i at i on , p i s the dens i ty of

the gas , v i s the ve loc i ty , pv is the m omentum , e i s the ene rgy pe r

uni t volume and p i s the p re s s ure . We have

1 4 8

( 4d ) e pa é- pv2

whe re e s . +— q , e . i s the int e rnal ene rgy pe r uni t mas s,

1 P( 4 6 ) E m E

whe re y i s a c ons tant , y 1 , and q is the ene rgy of format i on

whi ch c an be re leas e d through chemi c al re ac t i on ( see In the

p re s ent s e c t i on i t wi ll b e as sume d that part of q i s re le a s e d

ins t ant ane ous ly in an infini t e ly thin re ac t i on z one . Le t the sub

s c r ipt 0 re fe r t o unburne d ga s ga s whi ch ha s not ye t unde r

gone the c hemi c al re ac t i on ) and le t the sub s c rip t 1 re fe r t o burne d

gas . The unburne d gas i s on the right . We have

1 pl

8 ql yl

- I pl1

81 p

o

0 yO- I pO

0

Fo r the s ake of s imp li c i ty , we shall make he re the unre al i s t i c

a s s ump t i on 7 1 yo y . ( The c as e 7 1 Y Oi s mo re d i ffi cult only

be c au se of add i t i onal algeb ra . ) When y l y the re ac t i on c anyo

be e xothe rmi c re le as e ene rgy ) only i f q l go

Le t U be the ve loc i ty of the re ac t i on z one . Le t

C ons e rvat i on o f mas s and momentum i s exp re s s ed by

From the s e

De fine the func t i on H by

( Tl- TO)( p i

+po)

H sl- eo

C ons e rvat i on of ene rgy i s exp re s s e d by

H O

De fine A qO— q l

, (A O for an exothe rmi c pr oc e s s ) , and H2

iii;we find

2 2 2 2H O ( l - LL ) Tlp l ( l —

LL ) Topo“ 2“ A + LL ( Tl

2 2 2— pO( TO-u T

l )+ p

l( Tl

—u T

o) - 2u A

In the ( T l , p l ) p lane the l oc us of po int s whi ch c an be c onne c t e d t o

( To, po )by an infini t e ly thin c ombus t i on wave is a curve whi c h

re duc e s t o a hype rb ola when A i s independent of p and 1 . ( Se e

Figure l . ) The line s through ( T t ange nt t o H O are c al le d0) po)

the Rayle igh line s . The i r po int s o f tangency , S1

and SQ, are

c alle d the Chapman- Jougue t ( CJ ) p oint s . The p ort i on , p 1 p0

and

TI

TO, of the curve i s omi t t e d b e c aus e i t c or re s pond s t o

2 2powo

+ po pl

wl+ pl

re lat i ons one re adi ly de duc e s

pO— pl

https://www.forgottenbooks.com/join

unphys i c al event s in whi c h M2

O . The uppe r por t i on o f the c urve

c or re s pond s t o de t onat i ons ; the p or t i on ab ove S1

t o s tr ong de t ona

t i ons and the port i on b e low t o weak de t onat i ons . The lowe r part

of the curve c or re s pond s t o de f lagrat i ons .

The ve loc i ty and s t re ngth o f a s t rong de t onat i on are ent i re ly

de te rm ine d by the s t at e of the unburne d gas in front of the de t ona

t i on and one quant i ty b ehind the de t onat i on , j us t as in the c as e

wi th s hock s . Le t po, pO

, To, 5

0and v be give n , as we ll as pl

,0

and as sume the unburne d ga s li e s t o the right of the de t onat i on .

We have fr om ( 7 )

2

T Tl O

u p0+ pl

p l+u p

Ou po

+pl

p l- pog s ome algeb ra y i e ld s

P2 1- 1 1

+ l l( 9 ) M P

opo( 2 2 ( 7

I f A 0 thi s f ormula re duc e s t o the expre s s i on for M in a shock ,

a s given in [ 2 ] or M is re al i f [ p ] ( y - l ) p A 0 ; thi s c an0

be re ad i ly s e en t o hold in a s t r ong de t onat i on .

The s t at e s on the curve H 0 l oc ate d b e twe en the CJ p oint S1

and the l ine T To

in the s t at e behind a we ak de t onat i on i s ent i re ly de t e rmine d

c or re s pond t o we ak de tonat i ons . As de s c rib e d

by the ve loc i ty U of the de t onat i on and the s t at e in front of i t .

1 5 2

In fac t , a weak de t onat i on c annot oc cur and what d oe s happen is a

CJ de t onat i on fol lowe d by a rare fac t i on wave . Our next ob j e c t ive

i s t o de r ive an exp l i c i t c r i t e r i on for de te rmining whe the r a

de t onat i on Wi l l b e a s t rong de t onat i on or a CJ de t onat i on .

It i s s hown in [ 4 ] that at sl, l l c

lwhe re c

l J§ 5E7BIis the s ound s pe e d , i . e . , a CJ de t onat i on m ove s wi th re s pe c t t o the

burne d ga s wi th a ve loc i ty e qual t o the ve loc i ty o f s ound in the

burne d ga s. We now us e thi s fac t t o de t e rmine the dens ity pCJ’

ve loc i ty v and pre s s ure pCJ

b ehind a CJ de t onat i on .

CJ

From e quat i on s ( 4 ) and ( 5 ) one find s

and thus in a CJ de tonat i on

p-vp /T T l/pT

ITo

1 pl1 1 l 1

O T

( 1 0 ) — po) vfo

pl

E quat ing Tl

ob taine d from ( 8 ) to T1

in we find

2

T

pi+ po 2u

gA YTo

pi

2 2p 1

+u po

( pl+u p

o) p 1 1 p

0

S ome algeb ra re duc e s thi s e quati on t o

pi+~ 2 plb + c o

whe re

0, II - p

oA ( y - J' ) po

2 2C p

O+ 2u p

OpOA

a t riv ial c alculat i on shows that b2- c 0 i f y 1 and A 0 . Thus

pCJ

p 1- b -+Jb E - c

s ign i s mandat o ry s inc e a de t onat i on i s

Pl, pCJ pl

1 11

c an be ob t aine d f rom

_plwl’ and w “ C

l ’we f ind

The ve loc i ty U of the de t onat i on i s f ound fr om

po(Vo

' UCJ) “ M

whi ch yi e ld s UCJ

( povoi VPCJpCJ

)/pCJ’ and then

v depend s only on the s t at e of the unburne d ga s .

Suppo s e v1 ,

the ve loc i ty of the burne d ga s , i s g iven . I f

v1 .2 v

CJa CJ de t onat i on wi l l appe ar , f ol l owe d by a rare fac t i on

wave . I f v a CJ de t onat i on wi l l appe ar al one , and i f1

VCJ

v1

v a s t rong de t onat i on wi l l t ake p lac e .

CJ

If the unburne d ga s li e s t o the le f t of the burne d gas

analogous re lat i ons are f ound ; the only d if fe renc e l ie s in the

s igns of v , in par t i cular,

1 54

https://www.forgottenbooks.com/join

t i ons wi th e ne rgy de po s i t i on in the flow fi e ld are e quivalent t o

the exo the rm i c c ent e r s int roduc e d by Oppenhe im [3 ] and s e rve the

s ame purpo s e of ac c oun t ing fo r the dynami c al e ffe c t s of the exo

the rm i c re ac t i ons . The s e d i s c re t e exothe rm i c c ente r s c orre s pond

t o a phy s i c al re ali ty who s e ori gin c an be as c rib e d t o the f luc tua

t i ons t o the leve l s of chem i c al s pe c i e s

We c ons i de r he re the s imp le s t p o s s ib le de s c r ip t i on of a re

ac t ing gas ( s e e e . g .

( 1 3 a ) pt+ ( pV ) X O

( l ) 0

( D C ) et+ o

whe re , as b e f ore , p i s the dens i ty , v is the ve lo c i ty , e the ene rgy

pe r unit vo lume ,

( 1 3 d ) e pa pV

e i s the inte rnal ene rgy . In t hi s s e c t i on ,

1 p1 Z( B e ) 8

373 pq

whe re y i s a c ons t ant , y 1 , q i s the t o tal avai lab le b onding

ene rgy ( q O ) , and Z i s a p rog re s s parame te r f or the re ac t i on .

T p/p is the t empe rature , and A is the c oe ff i c ient of he at c on

duc t i on . Z i s as sume d t o s at i s fy the rate e quat i on

( B f ) gg — KZ , z ( o ) l

whe re

1 5 6

W l

l 0 H.

Ff)

r—3 l

l

Ti

.DA

1—3

TO is the ignit i on tempe rature and K

e quat i ons of the p re c e d ing s e c t i on are re c ove re d i f we se t x O,

Oi s the re ac t i on rate . The

q A , and K a ) . E quat i on ( l ) i s a re as onab le p r ot otype of the

vas tly more c omp lex e quat i ons whi c h de s c r ibe re al chem i c al

kine t i c s . Vi s c ous e ffe c t s have be en om i tt e d he re ; the i r inc lus i on

in the p re s ent c ont ext ha s l i tt le e ffe c t and pre s ent s li t t l e

di ffi culty . ( Thus , we as s ume he re a z e r o Prandt l numb e r . )

The app roximat i on of the di s s ipat i on te rm wi ll b e re legate d

t o a se parate f rac t i onal s t ep , whe re i t i s t o b e handle d by

s t raight forward finit e d i ffe renc e s . In v i ew of ( l3 e ) , and the

pe rfe c t ga s law T p/p ( in app r opr i at e uni t s ) , thi s frac t i onal

s t ep re qui re s me re ly the app roximat i on of

( 1 4 ) BtT ( y- 1 )T

The d i f fe renc ing of a he at c onduc t i on t e rm alone int roduc e s

negligib le nume ri c al d i s s ipati on . Seve ral more s ophi s ti c at e d

app roximat i on me thod s we re t ri e d , but d id not s e em t o b e worth

pursuing .

Al l that remains t o be d one i s t o de s c rib e the s oluti on o f

the Riemann p rob lem fo r e quat i ons ( 1 3 ) wi th A O . Thi s wi ll be

d one wi th the following s impli fying as s ump t i on : whateve r ene rgy

may be re leas e d during the t ime k/? in a port i on of the fluid i s

re leas e d ins tant ane ous ly . Thi s appr oximat i on i s we ll in the s pi ri t

of our me thod ( s inc e i t app roximate s Z by a p ie c ewi s e c ons tant

func t i on ) ; i t al s o ha s s ome phys i c al j us t i f i c at i on

Our goal i s t o s o lve e quat i ons ( 1 3 ) and the fo ll owing data

S£( p pg

’ p pg, v V

g’ Z Z

g) fo r x O

Sr( p pr , p p

r, v V

r’ Z Z

r) fo r x O

wi th k O . We b egin by a part i al revi ew of the c as e K 0 ( no0

chem i s t ry ; s e e The s o lut i on c ons i s t s of a ri ght

s t ate Sr’ a le ft s t ate S a mi d dle s t ate S

* ( p p* , vE,

s eparate d by wave s whi ch are e i the r rare fac t i ons or shocks . S*

i s

d ivi de d by the s lip line g% v*

int o two part s wi th po s s ib ly

di ffe ring value s of p, t o the right of the s lip l ine and p* zt op-X-r

i t s le ft . To de te rm ine v*

and p* we p r oc e e d as foll ow s : de fine

the quant i ty

( 1 5 )

I f the r ight wave i s a sho ck,

( 1 6 ) M —pr (

V - U — Ur)

whe re Ur

i s the ve l oc i ty of the right s hock . F rom the Rankine

Hugoni ot c ondi t i ons one ob t ains

( 1 7 a ) Mr

p*/p r 1

whe re

1 5 8

https://www.forgottenbooks.com/join

v+ lPi ck a s tart ing value pg ( or value s M3, M and then c omput e p*

(Qvv

M!+ l

, M2+ l

, q 0 us ing

«4; V V V V

( 23 a ) p ( uz- u

r

-+ pr/M

r

( 23 10 ) p21”

max

v+ 1 v+ 1( 23 0 ) M

r

=r / prpr ¢ ( pi /p

r)

( 23 d ) M)”

= Jp£ pg

MpXH/pg)

E quat i on ( 23 b ) i s ne e de d b e c aus e the re i s no guarant e e that in the

- 6c our s e of i t e rat i on p remains 0 . We us ually s e t 2

1lo The

i t e rat i on is s t oppe d when

v+ l v v+ l vmax M - Mr

M3

“ M2

82

( we usually pi cke d 62

lo_6) ; one then s e t s M

rM!

+ l, M

2 2+ l

,

+ 1and p* p:

To s tart thi s pro c e dure one nee d s ini t i al value s of e i the r Mr

and Mg( or The s t art ing pr oce dure sugge s t e d by Godunov

appear s t o be ine ffe c t ive , and be t te r re s ult s we re ob t aine d by

s e t t ing

0p*

We al s o ens ure d that the i t e rat i on was c ar rie d out at leas t twi c e ,

to avoi d spuri ous c onve rgenc e when pr

pg

.

As not e d by Godun ov, the i te rat i on may fai l t o c onve rge in

the p r e s enc e of a s t r ong rare fac t i on . Thi s p r ob lem c an be ove i c ome

1 6 0

by the fo llowing variant of Godunov 's pr oc e dure : If the i t e rat i on

has not c onve rge d aft e r L i te rat i ons ( we us ually s e t L

e quat i on ( 23 b ) is rep lac e d by

( 23b ) ' pi a max ( el , p — a ) p

wi th a al Q

“ If fur the r L i te rat i ons o c cur wi thout c onve rgenc e,

we re s e t a2

dl/2 . More gene rally , the pr ogram wa s wr i t t en in

such a way that if t he i t e rat i on fai l s t o c onve rge afte r 2L i te ra

t i ons ( 2 inte ge r ) , a i s re s e t t o

z“2

In p rac t i c e , the c as e s z 2 we re neve r enc ounte re d . The numb e r of

i t e rat i ons re qui re d os c i llate d be twe en 2 and 1 0 , exc ep t at a ve ry

few point s .

Once p* , Mr’ M are known , we have

2

( 24 ) v* ( p

g—pri- M

rUri

f rom the de fini t i ons of Mrand M

E

C ons ide r now the c a s e KO

O , ( k O ) ; the right and le ft

wave s may now b e CJ or s t rong de t onat i ons as we ll as shocks and

rare fac t i ons . The task at hand i s t o inc orpo rate the se pos s ib i li

t i e s int o the s olut i on of the Ri emann prob lem .

The s tate Srwi ll remain a c ons t ant s t a te ; v “

and pra re

fixed . The ene rgy in STmus t change at c ons tan t volume ( and thus

c an d o no wo rk ) . The c hange 6 2r

in Zr

c an b e f und by integ ra t ing

equa t i nS ( i3 r ) , ( 1 3 g ) , wi th z ( o) z,

a nd Z( k 2 ) zhi-oz

6Z” O . The new pre s s ure i s

( 2 5 ) prt fi p

rp + (v Qprr

( s e e e quat i on We w ri t e pnew

pr

-t opf, and dr op the s upe r

s c r i pt new . ( We shall ne e d the o ld Zr

again and thus re frain fr om

renam ing Zri- b z

r. ) S im i larly , 2

2c hange s t o z

£+- oz

£,

i s found us ing the obvi ous analogue of e quat i on

and a new pfl

In S*

the value s o f Z d i ffe r f r om the value s zr+- oz

r, 2

34- 6 2

3.

Le t Z* £

b e the value of Z t o the le ft of the s li p l ine and le t Z* r

b e the value of Z t o the ri ght of the s li p line . The d i ffe renc e in

ene rgy of format i on ac r os s the right wave i s A I! (Zr

and ac r os s the le ft wave i t i s Ag

(Z* £- (Z

£We shall

i t e rate on the value s Z* r

’ Ar, A

g“ In the fi r s t i t e rat i on , we

s e t Z* r

Zri- bz

r, Z* £

Zz

-t b zfi, and thus A

rA O , and c ar ry3

out the i te rat i ons When ( 23 ) has c onve rge d , a new p re s sure

p* i s given , and new dens i t i e s p* r , p* z

c an be found fr om e quat i ons

( l6 ) , ( 2 1 ) or the i s ent rop i c law . New tempe rature s T* r

T* z

are evaluat e d , e quat i on s ( l ) , ( 1 3 g ) are s o lve d , and

new value s Z* r

’ Ar’ A

gare found . If A

rO the ri ght wave i s

e i the r a s ho ck or a rare fac t i on , and i f Ar

O the right wave i s

e i the r a CJ de t onat i on fo l lowe d by a rare fac t i on or a s t rong

de t onat i on .

Le t v*

b e the ve loc i ty in S*

. G iven Ar’ A

g) we c an find the

ve loc i t ie s VCJr

’ VOJEb e hind po s s ib le CJ de t onat i ons on the right

and le ft ( e quat i on I f V*

.i v the right wave i s a CJCJr

de t onat i on fo llowe d by rare fac t i on , and i f V* .i v the right wave

CJr

i s a s t r ong de t onat i on . The CJ s t at e i s unaffe c te d by S* ( s inc e i t

de pend s only on Sr) and a s far as the Riemann s olut i on i s c onc e rne d

i t i s a fixe d s tate . If the r ight wave is a CJ de t onat i on , we re

de fine Mr

‘

1 62

https://www.forgottenbooks.com/join

4 — 2 8

i f r ight wave s t rong de t onat i on,

m m V + 1 N

Pr pr ¢ ( P* /p othe rwi s e ,

Jpgpz

i f le f t wave s t rong de t onat i on,

m m V + 1 ’V

( Pfl pg

¢ ( p* /p£) othe rw i s e .

The c omp lexi ty o f thi s i t e rat i on i s m o re apparent than re al I t is

s t oppe d when i t has c onve rge d , as b e fore . New value s of Z* r

’ Z* £

Ar, A

gare evaluat e d , and the i te rat i on is repe at e d ; thi s p roc e s s

i s s t oppe d when Ar’ A change by le s s than s ome p re de te rmine d 8

3fl

ove r two suc c e s s ive i t e rat i ons . It c an be re adi ly s e en that wi th

the p re s ent exp re s s i on fo r the ene rgy of format i on , at m o s t f our

i t e r at i ons on Ar’ A

gare eve r nee de d .

Onc e S*ha s b een de t e rm ine d , the s olut i on mus t b e s amp le d .

Le t P ( 8h , k/2 ) be the s amp le p o int , and p p ( P ) , p p ( P ) , e t c .

Four ba s i c c as e s are t o b e c ons ide re d

A ) P li e s t o the r ight o f the s lip line and the r ight wave i s

e i the r a s hock or a s t r ong de t onat i on ;

B ) P li e s t o the right o f the s li p line and the right wave i s

e i the r a rare fac t i on o r a CJ de t onat i on fol lowe d by a

rare fac t i on ;

c ) P 1 ie s t o the le f t of t he s lip line and the le ft wave i s

e i the r a shock or a s t rong de t onat i on , and

D ) P lie s t o the l e ft of the s lip l ine and the le ft wave i s

e i the r a rare fac t i on o r a CJ de t onat i on foll owe d by a

rare fac t i on .

l6u

Cas e A . The ve loc i ty Ur

of the shock or the s t rong de t onat i on c an

b e found from the re lat i onship

Mr

“

pr (Vr

- Ur) 5

dxHt

p pr , v Vr’ Z Z

r+ b z

r. I f P li e s t o the le ft of

atUr’ we

i f P li e s t o the ri ght of Urwe have the s amp le d value s p pr ,

have p p* r, P P* , V V

* , Z Z* r

°

Cas e B . C ons i de r fi r s t the c as e of a rare fac t i on wave . The rare

fac t i on is b ounde d on the ri ght by the line g% v -+ cr r ’

dxcr

,/ypr7pr , and on the le f t hy '

dtv*+- c

* r, whe re c

*c an b e

found by us ing the c ons t ancy of the Ri emann invariant

1 1Fr

2 0 ( Y ‘ l ) “ V*

e0r( 7 l ) V

r

If P l i e s t o the right of the rare fac t i on , p pr ’ p pr, v V

r’

Z Zr+- 6Z

r. I f P li e s t o the le ft of the rare fac t i on , p p* r

’

P P* , V v* , Z Z

r+- dz

r. I f P l i e s ins i de the rare fac t i on , we

e quate the s lope of the c harac t e ri s t i c v i- c to the s lope of

the l ine through the origin and P , ob t aining

v -t c 2 8 h/k

the c ons t ancy o f rr, the i s ent rop i c law pp

“

y c ons tant and the

de fini t i on c /7E75 yie ld p , v , and B . 2 zr4- 5 z

r

. If the wave

i s a CJ de t onat ion , are re plac e d eve rywhe re by

and Zins ide the fan and to le ft of i t s e qua ls Z'r

'

The c as e s C and D are mi rror image s of A and B , and wi ll not

b e de s c r ibe d in full .

Nume ri c al Re s ult s

We begin by pre s ent ing s ome re sult s for de t onat i on wave s wi th

0( KO

The s e re s ul t s ve r i fy the ac curacy o f

the p rogramm ing rathe r than the gene ral vali d i ty o f the me thod ,

ve ry large K

s inc e the s olut i ons of the c orre s ponding p rob lem s are an int rins i c

part of the Riemann p rob lem s olut i on rout ine .

To ob tain Tab le I , I s ta rt e d wi th a gas at re s t , p l , v O,

p l,

and at t O impo s e d impul s ive ly on the le ft b oundary c ondi

v 1 . 1 us e d h k/h 2 , K 1 0 0 0 , TO 0

q l and y The re sult i s a pe rfe c t s t r ong de t onat i on .

t i on v

In Tab le II a Chapman- Jougue t de t onat i on i s exhib i t e d .

h k/h 2 , KO

1 0 0 0 , To

q 1 2 and y m 1 1

The s o lut i on i s exhib i t e d at t 2 , n t/k 9 , i . e . n i s no t a

mult ip le of m and the s olut i on i s not at i t s mo s t ac curate . Thi s

c an be s e en fr om the p re s enc e of a fake c ons t ant s t at e ( for x

and whi ch was di s cus s e d in the s e c t i on ab out e rror s , and

whi ch i s mo s t like ly t o appe ar when n is no t a mul t ip le of m . The

la s t c olumn pre s ent s the ri ght Riemann invariant Frwhi ch i s of

c ours e c ons t ant b ehind the CJ fr ont . The c hemi c al t ime s c ale i s

not re s olve d on the g ri d,

and one should not i c e the small numb e r

of me sh point s re qui re d t o d i s play sharp variat i ons in all quant i

t i e s .

https://www.forgottenbooks.com/join

k/h

Tab le I I

1 68

1 0 0 0 , 1 1,

II

We now p re s ent s ome re sul t s for a pr ob lem whos e s olut i on is

not pr ogramme d int o the s olut i on algor i thm - a de flagrat i on wave

wi th fini te re ac t i on rat e . Fo r t O a gas at re s t li e s in x O ,

wi th p l,p l , ( v O ) , and Z l ; the le ft b oundary i s

maint aine d at z e ro ve loc i ty , V 0 . At t O the ga s in the fi r s t

c e ll t o the le ft is rai s e d t o a tempe rature T 2 , ( i . e . the

pre s sure is inc re as e d t o p The re su lt ing de flag rat i on wave

is ob s e rve d . It is known that the ve loc i ty of the wave i s

asymp t ot i c ally pr opor t i onal t o ,/ffig ( s e e e . g . p . thus ,

the wave d oe s not pr opagate unl e s s k 0 , a s one c an re adi ly ve r i fy

on the c ompute r . Thi s las t j us t i fi e s an e ar li e r as s e r t i on t o the

e ffe c t that when k O the wave is ind i s t ingui shab le from a s lip

line . The re sul t s in Tab le I II we re ob t aine d wi th h

k/h . 3 5 , To

KO

l , q 1 0 , y and m 1 1 . They are

pre s ente d at t nk . 273 , ( n q ) . One c an c le arly s e e the pre

curs or shock,

and the de f lagrat i on z one ( charac t e ri z e d by Z l )

in whi ch the dens i ty and p re s s ure de c re as e . The small numb e r of

me sh po int s should again be not i c e d .

. 3 5 , t nk

Tab le I I I

-273 , n

1 7 0

9 ,

https://www.forgottenbooks.com/join

A . A . Bor i s ov , Ac t a As t ronaut i c a , l , 9 0 9

L . M . C ohen , J . M . Shor t and A . K . Oppenhe im , C ombus t i on and

Flame,24 , 3 1 9

R . C ourant and K . O . F ri e d ri chs , Supe r s oni c Flow and Shock

Wave s , Inte r s c i enc e

Godunov,Mat . Sb ornik, 47 , 27 1

Lax , S IAM Rev iew , 1 1 , 7

( 1 977 )

R . D . Richtmye r and K . W . Mort on , Fini t e Di ffe renc e Me thod s

fo r Ini t i al Value Prob lem s , Inte r s c i en c e

F . A . Wi lli am s , C ombus t i on The ory , Add i s on - We s ley

A NUME RICAL STUDY OF CYLINDRICAL IMPLOS ION

Gary A . So d

C ourant Ins t i tut e of Mathemat i c al S c i ence sNew York Unive r s i ty

New York , New York 1 0 0 1 2

Ab s t rac t—

A nume ri c al p roc e dure is int roduce d t o s olve the oned imens i onal e quat i ons of ga s dynami c s for a cylindri c ally ors phe r i c ally symme t r i c fl ow . The me thod c ons i s t s of a j udi c i ousc omb inat i on of Glimm ‘

s me tho d and ope rat or s p li t t ing . The me thodis app lie d t o the pr ob lem of a c onve rg ing cylindr i c al shock .

Int roduc t i on

The one - d imens i onal e quat i ons f or an invi s c i d , non- heat c on

duc t ing , radially symme t ri c flow c an be wri t t en in the fo rm

( 1 ) Ut+ g(g) r

-w( u>

whe re

m/r

( 2 ) U a ( u) m9/p p and w( U ) ( a - l ) m

e/pr

whe re p i s the dens i ty , u i s the ve loc i ty,m pu i s the momentum,

p i s the p re s s ure , e i s t he ene rgy pe r un i t volume , t i s t ime ,

r i s the s pac e c oord inate of symme t ry , a i s a c ons tant whi ch i s 2

for cy lind r i c al symme t ry and 3 for s phe ri c al symme t ry , and the s ub

s c ri p t s re fe r to d i ffe rent i ati on . We may wri te

e =

fi + é pu2

whe re y i s the rat i o of s pe c i f i c he at s ( a c ons tant gre at e r than

The re are two maj o r p roblem s involve d in s olving the sy s tem

( 1 ) d i re c t ly . The fi r s t i s the s ingular nature ne ar the axi s

( r that i s , the re are s ingular te rm s p r opor t i onal t o l/r .

The s e c ond pr ob lem i s that the momentum e quat i on ( the s e c ond c omp o

nent e quat i on of c annot b e put in c ons e rvat i on fo rm .

The s e p rob lem s c aus e maj or d i ffi c ul t i e s near the axi s . The s e

are us ually ove rc ome by s ome ad hoc me thod such a s ext rapolat i on

( Payne Ano the r app r oach has b e en t o t re at thi s a s a pr ob lem

in C ar te s i an c oord inat e s in two s pac e d imens i ons ( Lap i dus

In the me thod de s c r ib e d b e l ow b o th of the s e p r ob lem s have

b e en c omp le te ly e l im inat ed . Thus the re i s not ne e d t o re s or t t o

any t r i cke ry in orde r t o s olve the sy s t em

Out l ine of the Me thod

The fi r s t s t e p in the pr ob lem i s t o us e the me thod known as

ope rat o r sp li t t ing t o rem ove the inhomogene ous t e rm s - W ( U ) f rom

the sy s t em Thus we s o lve the sy s t em

(A ) U,

CF ( U )

ro

whi ch re pre s ent s the one - dimens i onal equat i ons of ga s dynami c s in

Car t e s i an c oord inate s .

The me thod us e d t o s olve sy s tem ( 4 ) i s the random choi c e

me thod int r oduc e d by G limm ( 1 965 ) and deve lope d fo r hyd rodynami c s

by Chor in De tai l s of thi s me tho d w i ll be given in the next

s e c t i on , fo r c omp le t ene s s .

1 714

https://www.forgottenbooks.com/join

Se e F igure 1 . Le t fin

be an e quidis t ribut ed rand om var iab le whi c h

1 1is g iven by t he Leb e s gue meas ure on the int e rval

Q ’ 2 J' De fine

m n+ l/2 l( 7 ) l <<1 + fi nM

T :

Se e F igure 2 .

At e ach t ime s t e p , the s olut i on i s appr oximat e d by a pi e c e

wi s e c ons t ant func t i on . The s olut i on i s then advanc e d ln t ime

exac t ly and the new value s are s amp le d . The me thod de pend s on

s olving the Riemann p rob l em exac t ly and inexpens ive ly .

Chorin ( 1 976 ) ( s ee al s o S od ( 1 976 ) and ( l97 8 a ) ) m odi fie d an

i t e rat ive me thod due t o Godunov ( 1 959 ) whi ch wi ll now be de s c r ib e d .

C ons i de r the sy s t em ( 4 ) wi th the ini t i al dat a

SB

r O

U ( r , o )

SI“

2 ( pr : ura p

r) l

" O

The s olut i on at lat e r t ime s look s like Figure 3 , whe r e S1

and 82

are e i the r a sho ck or c ente re d rare fac t i on wave . The re gi on S*

i s

a s te ady s t at e . The line s 21

and 22

are s e parat ing the s t at e s .

The c ontac t s urface u*

s ep arate s the re gi on int o two part s

wi th po s s ib ly d i ffe re nt value s of p* , but e qual value s of u*

and p*

Us ing thi s i t e rat ive me thod we f i r s t evaluat e p* in the

s t ate S*

. De f ine the quant i ty

( 9 ) M

I f the le f t wave is a shock,

us ing the j ump c ondit i on [ pul,

we ob tain

1 7 6

1=M +J ) A t

l=nA 1

{1 (x— I) A r ( l IA ? ( t + l) A r

Figure Se quenc e of Riemann p rob lem s on gri d .

I= nA l

,A r ( c+ i) A r

Figure 2 . Sampl ing pr oc e dure for G limm 's s cheme

.

whe re U3is the ve loc i ty o f the le ft shock and p* i s the dens i ty

in the por t i on of S*

ad j oining the le ft shock . S im i lar ly , de f ine

the quant i ty

( 1 1 )

I f the ri ght wave i s a s hock , us ing the j ump c ond i t i on Ur[ p] = [ pu ] ,

we ob t ain

Mr

Ur)

whe re Ur

i s the ve loc i ty of the r ight s ho ck and p* i s dens i ty

in the p ort i on o f S*

ad j o ining the right s ho ck .

In e i the r c as e or ( 1 0 ) fo r M and ( 1 1 ) or for Mr)

A

we ob t ain

Jpfipfl

= / p”P ( Pa /pg )

1- 1 l — x

x 1

Up on e l iminat i on of u*

from ( 9 ) and ( 1 1 ) we ob tain

https://www.forgottenbooks.com/join

f ir s t orde r ac c urat e s o the re i s no re as on for us ing a high o rde r

me thod fo r s o lving the sy s tem of ord inary di ffe rent i al e quat i ons .

S inc e th i s sy s t em ( 5 ) i s s olve d only at inte r i or p o int s and

the s cheme ( 1 8 ) d oe s not re qui re value s at r O,

the s ingulari ty

at the axi s i s e lim inate d .

Boundary c ondi t i ons ne e d only b e app li e d t o the sy s tem (4 )

s inc e the sy s t em of ord inary d i ffe re nt i al e qua t i ons ( 5 ) us e only

inte ri or po int s . S o that wi th the p ro c e dure de s c rib e d by Cho rin

( 1 976 ) the b oundary c ondi t i on at the axi s ( r O ) i s re adi lyII

handle d . The b oundary c ondi t i on i s imp o s e d on the gri d p oint

c lo s e s t t o r 0 , s ay loA r . A fake le f t s t at e i s c re ate d at

l( lo

by s e t t ing

re fle c t whi ch

Ini t ial ly , a cy lind r i c al di aphragm of radius r s ep arate s two0

un i fo rm regi ons of g a s at re s t as in a shock tube wi th the oute r

1 8 0

P p10

10+ l/2

u .- u o

lo— l/2 i

O+ l/2

P P10

iO+ l/2

rare fac t i on wave w i l l

Radius

Figure A . Flow pat te rn for c onve rg ing cy l indri c al shock .

pre s sure and dens i ty be ing large r than the inne r one s . Af te r the

d i aphragm i s rup ture d ( t a s ho ck wave i s c re at e d and t rave l s

into the low p re s s ure re gi on fo ll owe d by a c ont ac t di s c ont inuity .

A rare fac t i on wave t rave l s int o the high p re s s ure re g i on . Se e

F igure 4 .

It i s known that a cy l ind r i c al s hoc k wave in a c omp re s s ib le

f luid inc re as e s in s t rength as i t c onve rge s t oward the axi s . Thi s

c an be s e en expe r iment ally in Pe rry and Kant r owi t z

In the examp le g iven be l ow the p re s s ure and the dens i ty in

the inne r re gi on we re s e t e qual t o and the p re s s ure and dens i ty

in the oute r re gi on we re s e t e qual t o A . O . Thi s wi l l pr oduc e a

s hock wi th ini t i al s t rength of a c ontac t di s c ont inui ty and a

rare fac t i on wave . We t ook A r The t ime s t ep A t i s c hos en

s o that the C ourant - Fri e d ri chs — Lewy c ondi t i on i s s at i s fi ed , i . e .

A tmax ( hl fi t c )z ? i

1

whe re c i s the l oc al s ound s p ee d .

In F igure 5 the p re s s ure di s t r ibut i on i s di s p lay e d at t ime

int e rval s of The shock app e ar s as a rap i d var i at ion in p

whi ch i s c omp le te ly sharp , i . e . the numbe r of z one s ove r whi ch thi s

variat ion t ake s p lac e is z e r o . As t ime inc re as e s the s hock pr opa

gate s t oward the axi s . I t i s ob s e rve d that the s t rength of the

shock inc re as e s wi th t ime . Afte r the pas s age of the s hock , the

p re s s ure b ehind the shock inc re as e s . When the shock arrive s at the

axi s i t i s re fle c te d and ri s e s t o a large but fini t e value and a

d ive rging sho ck appe ar s . It is al s o ob s e rve d that the p re s sure at

a given p oint be hind the re f le c te d sho ck de c re as e s wi th t ime .

1 8 2

https://www.forgottenbooks.com/join

In F igure 6 the ve l oc i ty o f the gas i s di s p laye d . The

b ehavi or i s s im i lar t o that of the p re s s ure exc e p t that the c on

ve rg ing s hock de c re as e s the ve l oc i ty from z e r o t o a negat ive value .

When the shoc k i s re f le c te d f rom the ax i s , the d ive rging sho ck has

the e ffe c t of p roduc ing a small p o s i t ive ( outward ) ve l o c i ty . As

in the c as e of the p re s s ure p r ofi le , at a given p oint b ehind the

c onve rg ing s hock the ve lo c i ty inc re as e s wi th t ime and behind a

dive rg ing shock the ve l oc i ty de c re as e s wi th t ime .

The dens i ty and ene rgy p r ofi le s are di s p laye d in Figure s 7

and 8 re s p e c t ive ly . The b as i c p rope r t ie s of the shock are s imi lar

t o tho s e of the p re s s ure di s t r ibut i on , exc ep t that the ris e in

dens i ty ac r o s s the sho ck is smalle r due t o a t empe rature inc re as e .

In the dens i ty and ene rgy p r ofi le s a c ontac t d i s c on t inuity appe ar s .

It i s a re s ul t of us ing G l imm ‘s s c heme that the c ontac t di s

c ont inui ty ( as we ll as the s hock wave ) i s c omple te ly s harp . The

c ont ac t d i s c ont inuity p r opagate s t oward the axi s b ehind the c on

ve rging shock and i s t rave r s e d by the re f le c t e d ( outg oing ) shock .

In Figure 9 the dens i ty p r ofi le whe re the c ont ac t di s

c ont inui ty and the re f le c te d shock wave have c r os s e d . For a poly

t rop i c gas wi th the s ame value s of y ,highe r s ound s p ee d s c o rre

spond t o highe r dens i t i e s ( C ourant and Fr ie dr i chs , The

inte rac t i on of a d ive rg ing s hock wave and a c ont ac t di s c ont inui ty

p ropagat ing t oward the axi s re s ult s in a re fle c t e d ( c onve rging )

shock ( re p re s ent e d by a c ontac t d i s c ont inui ty p r opagat ing

t oward the axi s ( repre s ente d by ) , and a t ransmi t te d ( d ive rging )

shock ( re p re s ente d by

1 84

0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 0

Figure 8 . E ne rgy pr ofi le s at t ime inte rvals of

Figure 9 . Dens i ty pro fi le afte r inte rac t ion of dive rginshoc k and c ont ac t di s c ont inuity at t ime t ==o.

In gene ral the ove ral l t rend of the re s ult s agre e wi th tho s e

o f Ab a rb ane l and Goldb e rg Lapi dus and Payne

The re is ,howe ve r , one maj o r d i ffe renc e , the t ime at whi ch the

s hock re ache s the ax 1 s . Our me thod is in agreement wi th the me thod

o f Ab a rb ane l and G oldbe rg . Howeve r , Wi th the me thod s o f Lap i dus

and Payne , the s hock re ache s the axi s s oone r .

It s hould b e no te d that as a re s ult of the rand omne s s of

Glimm 's me thod , at a g iven t ime , the p o s i t i on of the s ho ck or c on

t ac t d i s c ont inui ty may not b e exac t . Ye t on the ave rage the i r

p o s i t i on s are exac t .

Wi th the thre e othe r me thod s us e d in thi s c ompar i s on , the

s hock and c ont ac t d i s c ont inui ty are sme are d . The sme ar ing of the

s hock i s le s s d ramat i c . The c ontac t di s c ont inui ty ob t aine d by

Payne 's me thod i s almo s t imm e d i at e ly sme are d t o s uch a degre e that

i t i s b are ly vi s ib le . Howeve r , our t e c hni que p r oduc e s pe r fe c t ly

s harp s ho cks and c ont ac t d i s c ont inui t ie s .

As di s c us s e d ab ove , the int e rac t i on o f the re f le c t e d shock

and the c ont ac t d i s c ont inui ty wi l l p roduc e a c ont ac t di s c ont inui ty ,

a t ransmi t t al shock and re f le c t e d s hock . The re fle c te d s hock i s

p roduc e d by our te chni que ( s e e Figure Howeve r , the re fle c t ed

shock i s no t p r oduc e d by the me thods of Ab a rna e l and G oldb e rg ,

Lapi dus and Payne .

C onc lus i on s

Thi s me thod re duc e s the prob lem of s olving the one — d imens i onal

e quat i ons of gas dynami c s f or a cy lindr i c al ly or sphe ri c ally

https://www.forgottenbooks.com/join

flow . Howeve r , thi s c an y ie ld imp ort ant informat i on c onc e rning

the re lat ive e ffe c t s on the flow fi e ld of valve s i ze , swi rl rate s ,

pi s ton and he ad ge ome t ry , engine s pe e ds .

Thi s wo rk was supp ort e d in p art by Nat i onal Sc ienc e Foundat i on ,

Grant MC S76 - 07 03 9 and the U . S . E ne rgy Re s earch and Deve lopmentA dm ini s t rat i on unde r C ont rac t W— 740 5 - E ng- 4 8 .

Re fe renc e s

Ab a rb ane l , S . , and Goldb e rg , M 1 97 2 ,

" Nume ri c al S olut i on ofQuas i - C ons e rvat ive Hype rb ol i c Sy s t em s the Cy lindri c al ShockProb lem "

, J . C omp . Phy s . 1 0 , 1 .

Chorin , A . J ., 1 976 ,

" Random Choi c e S olut i on of Hype rb ol i cSy s tem s J . C omp . Phys . 2 2 , 5 1 7 .

1 977 ,

" Random Choi c e Me tho d s wi th Appli c at i on s t oRe ac ting Ga s Flow J . C omp . Phy s . 2 5 , 2 53 .

Sho ck Wave s , Int e r s c i enc e , New Y ork .

Glimm, J ., 1 965 ,

" Solut i on s in the Large fo r Nonlinear Hype rb oli cSy s tem s of E quat i on s " , C omm . Pure Appl . Math . 1 8 , 697 .

Godunov , S . K ., 1 959 ,

" Fini te Di ffe renc e Me thod s fo r Nume r i c alC omput at i on of Di s c ont inuous S olut i on s of the E quat i ons of Flui dDynami c s " , Mat . Sb ornik , 47 , 27 1 .

Lapi dus , A . , 1 97 1 ,

" C omput at i on o f Radi ally Symme t ri c Shocke dFlows J . C omp . Phy s . 8 , 1 06 .

Payne , R . B . , 1 956 ,

"A Nume r i c al Me thod for C onve rging Cy l ind ri c al

Shock J . Flui d Me ch . 2 , 1 8 5 .

Pe r ry , R . W . , and Kant r owi t z , A . , 1 95 1" The Produc t i on and

S t ab i l i ty of C onve rging Shock Wave s J . Appl . Phy s . 2 2 , 87 8 .

Sod , G . A . , 1 976 ,

" The C ompute r Implement at i on o f G limm's Me thod

Lawrenc e Live rmore Lab orat ory Repor t UCID- 1 7 2 5 2 .

l97 8 a ,

”A Survey of Seve ral F init e Di ffe rence Me thod s

for Sy s tem s of Nonl ine ar Hype rb oli c C ons e rvat i on Laws " , J . C omp .

Phy s ., 27 , l .

1 97 8 b ,

"A Nume ri c al Me thod for Ax i symme t ri c Flows wi th

Applic at i on t o Inte rnal C ombus t i on E ngine s " , J . C omp . Phy s . ,

t o appe ar .

l97sc , A Nume ri c al Mode l o f Uns te ady C ombus t i onPhenomena , t o appear .

C OMBUSTION INSTABILITY

Samue l Burs te in

C ourant Ins t i tut e of Mathemat i c al S c ienc e sNew Y ork Unive r s i ty

New Y ork , New Y ork 1 0 0 1 2

1 THE RMALLY INDUC E D VIBRATIONS

I t has b e en known for we ll ove r one hund re d ye ar s that the re

is ,unde r s ui t ab le c ond i t i on s , a s t rong int e rac t i on b e twe en s ound

wave s and flame s . Ray le igh de s c r ib e s a s imp le expe r iment by whi ch

a hi gh fre quency s ound,when app l ie d t o the po int o f e fflux of a

pre s suri ze d gas j e t t ing int o a quie s c ent envi ronment , c aus e s the

c ombus t i on p r oc e s s t o inc re as e in intens i ty . The flame roar s and

the t ot al d i s t ance re qui re d f or c omp le t e burning t o t ake p lac e i s

s ub s t ant i al ly re duce d s o that the e ff lux re gi on , al s o c alle d a

" p rehe at " z one,

i s whe re di f fus i on of heat and mo le cule s of the

inte rme di at e p roduc t s of re ac t i on t ake p lac e and dominat e all othe r

pr oc e s s e s .

Fr om thi s s imp le ob s e rvat i on ( and al s o fr om j e t engine

de s ign ) i t i s s t r ong ly s uspe c te d that a s im i lar , although mo re

c omplex,

pr oc e s s oc c ur s in the int e rnal c ombus t i on me chani sm of a

l i qui d pr ope llant rocke t m o t o r . In such a mot or l i qui d j e t s of

fue l and oxi d i z e r are d i s charge d fr om an inj e c t or he ad at the b as e

of a c ombus t i on chamb e r . The l oc at i on o f the re gi on of intens e

c ombus t i on de pend s not only on the de s ign parame t e r s of the inje c

t i on sy s t em,

inc lud ing fue l and oxi d i z e r pr ope r t i e s , but upon the

c omplex li qui d and ga s phas e mixing p r oc e s s e s oc c ur r ing in the

https://www.forgottenbooks.com/join

d rop le t fi e ld , the fo rme r of whi ch i s gene rate d f r om the burning

of the fue l d r op le t fi e ld . The dr ople t s are p roduc e d by b re akup

of the s p ray s of li qui d fue l and oxi d i z e r j e t s ; the d r op le t burn ing

me chani sm is as sume d t o b e de s c r ibe d by an evapo rat i on rat e

c ont roll ing pr oc e s s whi ch i s s lowe r than the chem i c al kine t i c

pr oc e s s by a s igni fi c ant tim e s c ale .

In add i t i on t o l i qui d pr ope l lant ro cke t m ot or s , the me thod

de s c r ib e d in thi s pape r c an b e us e d as a b as i s for the analy s i s of

d i re c t fue l in j e c t e d eng ine s , i . e . , d ie s e l or s t rat i fie d charge

engine s .

2 . FORMULATION OF THE RMAL FORC ING FUNCTION

Le t the prope r t i e s of the dr op le t b e denote d by w*

The dr op le t ve l o c i ty in the x2, x

3di re c t i on i s u

*and v

*and the

li qui d mas s,m The fue l d r op le t i s then c omp le t e ly s pe c i f i e d i f

~x»

e the int e rnal ene rgy i s known ; e cVT C? i s the s pe c i fi c

he at at c ons t ant volume o f the dr op le t . The c on s e rvat i on laws for

the fue l d r op le t c an be wri t ten in the c onveni ent form

w ) T + a w m > = o

s ub j e c t t o the ini t i al c ond i t i on

wO

He re we u se a Lagrangi an re pre s ent at i on s o that for an

annular c oord inate sy s tem and the part i c le

de r ivat ive is

at“ ‘

5 ? FK'

E P 5 2

The value of r*is the rad ius of the annular d omain ; i t i s t aken

as a c ons t ant and fo r c onveni enc e r*

1 . The inhomogene ous t e rm

having de pendence on drop le t and pr oduc t ga s pr ope rt i e s ,

is given by

s 4; fp

The rate of evaporat i on of the d r ople t is m* whi le the ae rodynami c

drag fo r ce s ac t ing on the d r ople t in the 8 and z d i re c t i on s are f6

and fZ

re s pe c t ive ly .

The inte rnal d r ople t t empe rature , T*

, i s c omput e d from the

integrate d C laus ius - C lapey ron e quat i on

( 5 ) r' 1

He re p* and T

’c o r re s pond t o the pre s s ure and tempe rature of the

d r op le t at the c ri t i c al point whi le the gas pre s s ure p p ( y - l ) e

i s t aken to be a func t i on of the c ombus t i on gas dens i ty p and

inte rnal ene rgy e ; the c ons t ant a i s the negat ive re c i proc al of the

vapor pre s s ure e qui lib rium c urve when the natural logari thm of the

vapor p re s sure is plot ted agains t the re c i p roc al of the va por

tempe rature .

1 93

The d rag fo rc e s are as s ume d t o b e de s c r ib e d by S toke s d rag

laws s o that f9

and fZ

are p roport i onal t o the s quare Of the re la

t ive ve loc i ty d i ffe renc e b e twe en the c ombus t i on ga s and l i qui d

dr op le t ,

The p rop ort i onal i ty fac t or i s the c oe f fi c i ent of d rag CD

and

a a ( t ) i s the d rop le t radi us whi ch c an b e g iven in te rm s of the

dr op le t mas s m* The d rag c oe ffi c i ent de pend s on the

d rop le t Reynolds numb e r Re* through

( 7 ) cD

In orde r t o c ompute the t ime rate of change of m* m* is

s pe c i fi e d thr ough an evap orat i on law given by

( 8 )“75“

+ 1 JRe*375 )

P P

wi th the c ombus t i on gas Prand t l numbe r de fine d by Pr Cpu/k . The

di ffus i on of the fue l vapo r f r om the s phe r i c al dr op le t i s re fle c te d

by the s pe c i fi c he at at c ons t ant p re s sure , Cp

, of vapo r and the rmal

c onduc t ivi ty,k , of fue l vapo r . The loc al Reynold s numb e r is

de fine d in te rm s of the d i ffe renc e of the magni tude s q of the l oc al

ga s and d rop le t ve loc i t i e s , i . e . Re*

2 pa The c ons tant

fl , ob t aine d empi ri c ally,

i s and p0is the ini t i al uni form

pre s s ure of und i s turb e d f low .

1 94

https://www.forgottenbooks.com/join

He re we have s e t the s ourc e te rm t o z e ro unde r the s impl i

fying as sumpt i ons that the dr op le t s wi ll not c o lli de wi th each

othe r ( di lute sp ray ) nor wi th the c hambe r wall . I t i s al s o as sume d

that the dr op s wi ll no t b e c re ate d by b re aking thr ough ae rodynamic

she ar for c e s no r b e c re at e d by nuc le at i on pr o c e s s e s .

Then M*

i s given by

M* 4v p*

a2Nad a du

*

Be fore we c an wri t e down the final form of our c ons e rvat i on

law we mus t de s c r ib e how t o pr oduc e a supe r s oni c out f low c ondi t i on

whi ch s imulat e s the s t ate of affai r s in a c onve rging - d ive rging

noz z le at t ache d t o the t ai l end o f the c ombus t i on chambe r .

Al though thi s is not the us ual p r oc e dure , i t turns out t o b e c on

venient . We as s ume that the re i s a c onve rging - dive rging duc t

p lac e d imme d i at e ly aft e r the uni form annular chambe r . In thi s duc t

we as sume that the rat e of change of flui d p r ope rt i e s normal t o the

s t re amline di re c t i on i s smal l c ompare d t o the rat e of c hange of

flui d p rope r t ie s along s t re aml ine s . If , in the duc t , we all ow fo r

a vari ab le c r o s s - s e c t i onal are a A whi ch de pend s only on the axi al

di s t anc e z , then we would expe c t small e rr or s in c omput ing s t re am

prope r t i e s i f d Ln A/d z i s small c ompare d wi th uni ty .

We pre s c rib e the s che dule of area vari at i on in the axi al

di re c t i on through

2l i— a

l (z — z

o) + d

2 ( z- zo)

1 9 6

whe re z is the t e rminal axi al po s i t i on of the c omb us t i on chambe r0

wi th rad ius r* l and are a A

O. The c ons t ant s a

land ( 1

2are cho s en

s o that

dA.

dA

E Z< O ’ Z

O< Z < Z

t’ z

t< Z < z

L

The t ot al length o f the s imulat e d noz z le i s 2L

and ztis the

po s i t i on of minimum are a , the throat , o f the noz z le . The only

c ondi t i on impos e d on the choi c e o f A ( z ) i s the re qui rement that the

pre s c ribe d s te ady s t at e fl ow , ( de s i gn flow ) b e shock- fre e .

Thi s is achi eve d by allowing the loc al Mach numbe r,M , at

z ztt o be uni ty for the asympt o t i c ally s te ady p rob lem . The

othe r c ond i t i on fixing the two c oe ffi c ient s s pe c i fying the are a

v ari at i on i s de t e rm ine d by p rovi ding for a large enough are a rat i o

be tween the throat and the p oint z 2L

s o that

( 1 0 ) 1

Thi s is the b oundary c ondi t i on re qui re d s o that c harac te r i s t i c

s urface s are po int ing int o the b oundary fr om the inte ri or of the

flow . Henc e,in the z d i re c t i on the thre e c harac te ri s t i c s V + c ,

v- c are all po s i t ive .

The d i ffe rent i al sy s tem , E q . c an now b e wri t ten in

d ive rgenc e fo rm

( 1 1 ) w

wi th the ve c t or B given by

1 97

unknowns w , axi al and the t angent i al

PV Pu

pu2+ p pvu

pvu pv2+ p

( E + p ) u

In E q . ( 1 2 ) we emphas i z e the de pendenc e of the e ne rgy and

mas s s ourc e s 8 on the int e rac t i on of the c ombus t i on gas wi th

p rope r t ie s w and d rop le t fi e ld wi th pr ope rt i e s w*

.

E quat i on ( 1 0 ) i s e quivalent t o our p re s c r ip t i on of ext rapo la

t ing from the inte ri or t o the b oundary 2 z the c ons e rvat i onL

vari ab le s w vi a the forward di ffe rence appr oximat i on

D+w O

The ne c e s s i ty for int roduc ing a no z z le int o the c alc ulat i on ,

even though only a s tudy o f pr oc e s s e s in the ne ighb orhood of the

inj e c t or fac e of the c omb us t i on chambe r i s de s i re d , s t ems fr om the

inab i l i ty t o de s c r ib e the c or re c t nonline ar t ime de pendent down

s t re am pre s sure leve l ln the c ombus t i on chambe r . Phy s i c al ly thi s

pre s s ure i s de t e rm ine d by flow b e ing choke d in the ne ighb o rhood of

the m inimum are a , whe re the Mach numbe r i s uni ty ; the f l ow then

ac c e le rate s t o a supe r s oni c s t ate downs t re am of the throat . At the

https://www.forgottenbooks.com/join

l ~n+ lvn+ 1

vn A t

( on

- G9

G1 J HAG 1 + I , J 1 — 1 , J l,

l i12 M 3 + 2

~n+ 1 ~n+ 1 A t nG l 1

G. 1 1 ) H

1 , 3- 1

+ 1 + 1 + 1 ~n+ l

'* fl+1

+1 $1 1 . 1 3

1

1+1

H. 1

l “

g2 3 51 l -

2'

, J gl -

n

A t 1 N n+ 1 ~n+ l ~n+ l[4

' ( B+1

B.

+1 1 1 +fl 11

21 1 ’

2’ J 2

~n+ l l n n

1( Bi+ l , j

Bl

l2’ J

2

w + 1 + 1H (V

pH? j

H (Vn

l 1 ) HH

1 l e t c .

“2 M “ ?

Ob t ain the fi r s t i t e r at e t o w*

, i . e ., by s olving E q .

( 3 ) us ing the analogue of sy s t em ( 1 5 ) and the m odi f ie d E ule r

me thod . The s e c ond i t e rat e t o w is now c ompute d us ing ( 1 5 ) and

Thi s p r oc e s s i s c ont inue d unt i l c onve rgence of w and w* i s

ac hieve d . The asympt ot i c l im i t of nA t , thinking of t ime as an

i te rat i on c ount e r , de fine s the s e lf - c on s i s t ent s t e ady s t ate

w ( nA t ) w

( DA L )0 /

To c omput e the evapo rat i on and c ombus t i on pr oc e s s , at any

point in the c ombus t i on chamb e r , one ne e d s t o ke e p t rack

of the prev i ous hi s t ory of the dr op le t . Sinc e a Lagrang i an

2 0 0

re pre s entat i on o f the dr ople t fi e ld has b e en ad opte d,e ach drop is

t racke d from it s point of in j e c t i on int o the c ombus t or 0 , t90,

O)

t o the the e lap s e d t ime of fl ight of the drople t

i s t*- t t the t ime of dr ople t inj e c t i on .

0’

0

We have as sumed a d i lute s p ray appr oximat i on whi c h me ans that

d rople t s d o not inte rac t . Al s o , a s a re s ul t of thi s app roximat i on

c an be evaluate d at E ule r i an me sh point s

of the sy s tem ( 1 5 ) and by inte rpo lat i on from

the Lag rangi an me sh up on whi c h i s de fine d . The

t racking p roc e s s c e a s e s i f

a ( t* — t . 1 a ( t

o)

0)

s inc e the mas s as s oc i ate d wi th the dr op would b e le s s than one

thous andth the orig inal d rop le t mas s ; fo r the s e c alc ulat i ons the

ini t i al me an d rople t rad ius wa s a ( to) 5 0 mi c r ons .

Wi th the s te ady s t ate e s t ab li she d , the flow fi e ld i s pe r

turbed t o ob s e rve the mode s of re s onanc e e s t ab li she d and maint ained

in the annular c ombus t i on chamb e r . Le t pObe the s t e ady s t ate

pre s sure ; the n the pe rturb e d pre s sure p' i s

I( 1 6 ) p p0+ p

lpo( l +- A S in 9 S in Z )

whe re we s c ale 8 and z by the func t i ons

a ( e ) el

< e < 92

Z( z ) s-r au

21

z z2

s o that the pe rturbation c an be placed at an arb i t rary pos i t ion in

the annular chambe r . Howeve r , j us t as Ray le igh de s c r ibe d the

import anc e of the plac ement o f the s ound s ourc e ne ar the e fflux

p oin t o f the ga s j e t , s o d o we ob s e rve that the pe rturb at i on mus t

b e p lac e d in s ome ne ighb o rho od of the po int o f inj e c t i on of the

d rops,

z 0 , t o have an app re c i ab le e ffe c t on the flow fi e ld . The

ampli tude of the pe rturb at i on , A , i s taken t o be p r opo r t i onal t o

the t o t al ene rgy of the c omb us t i on gas in the annular chamb e r ,

usual ly a few pe rc ent of the s t e ady s t ate chamb e r ene rgy . Fo r

the s e c alc ulat i ons , a value of A i s cho s en s o that the t ot al pe r

turb e d ene rgy in the re c t angle A 8 , A 2 c en te re d ab out

i s e qual t o four pe r c ent o f the ene rgy o f the c ombus t i on

gas .

We found that the t rans ien t s ob t aine d by thi s d i s turb anc e

we re s o s eve re that s t rong s hock wave s we re gene rate d . The d i f fe r

enc e s cheme E q s . ( 1 5 ) and d i d not remain s t ab le in the

pre s enc e of the s e s t e ep grad i ent s s o that a sm oothing ope rat or was

re qui re d . A two s t ep ope rat o r was us e d . Le t D deno te the b ack

ward d i ffe rence ope rat o r

D wm+ l

Wm+ l

" wm

then

A t( 1 7 )

-( ID-

um+ 1 | D

wm+ 1

)

whe re u i s the ve lo c i ty in the m- th d i re c t i on , m i s the s t ep s i z e

in that d i re c t i on and k i s a c on s t ant . E quat i on ( 1 7 ) is f i rs t

appli e d t o the s olut i on w in m- c oord inate d i re c t i on t o ob t ain a

temporary value w ; re plac ing w wi th w , E q . ( 1 7 ) i s then app lie d

onc e more t o yi e ld the final s olut i on . I t was al s o found that E q .

2 0 2

https://www.forgottenbooks.com/join

RE FE RE NCE S

l Bur s te in , S . , Nonline ar Time Dependent Prob lem s in F luid

Dynami c s , AGARD Le c ture Se r ie s No . 64 on Advanc e in Nume ri c al

Flui d Dynami c s .

2 . Wi ll i am s, P .A .

, C ombus t i on The o ry , Add i s on— We s ley , 1 965 .

2 04

0 - 1 0

Figure l ( a ) . Pre s s ure hi s t o ry at thre e po s i t i on s inannular re s onant c av i ty for a sp inning wave .

Figure l ( b ) . Dr iving ene rgy s ource d i s t ribut i on at t 3 .

L2 L6 2 0 2 4 2B

Th m

Figure 2 . Att enuate d p re s sure hi s t ory at the s ame thre epo s i t i ons as in Fig . l ( a ) in an annular re s onantc avi ty but wi th two e qually s pac e d b affle sins e r te d ne ar the inj e c t or .

https://www.forgottenbooks.com/join

he at ing of the s oli d fue l . Howeve r , natural or for ce d c onve c t i on

i s allowe d in a d i re c t i on opp o s i t e to that of f lame propagat i on .

Typi c al ly , the fue l might be a po lyme r . Thi s polyme r wi ll

b e as sume d t o gas i fy d i re c t ly ; that i s , no molt en laye r exi s t s .

The s pe c i al c as e wi ll be c on s i de re d in thi s le c ture whe reby

the oxi dat i on i s an exothe rm i c surfac e re ac t i on . Mos t polyme r s ,

of c our s e , would not burn in thi s manne r but rathe r in the ga s

phas e . Howeve r , thi s as sumpt i on re duc e s the mathemat i c al c omplex

i ty in two ways : ( 1 ) the numbe r o f gove rning par t i al di ffe rent i al

e quat i ons i s re duc e d s inc e the fue l s p e c ie s n o longe r exi s t s in

the vapo r phas e and as we shall late r s e e , the two - d imens i onal

pr ob lem may b e re duc e d t o a one — d imen s i onal p r ob lem .

The c a s e of s t e ady pr opagat i on wi ll b e c ons i de re d whe re the

s pre ading rate i s t o b e de te rm ine d as a func t i on of amb ient t em

pe ra ture ,p re s s ure , and oxi d i z e r c onc ent rat i on , t ransp or t p r ope r

t i e s,

and the rm ochem i c al pr ope r t i e s . The ve loc i ty of the inc oming

flow due t o f orc e d or natural c onve c t i on shall b e as sume d t o b e

known .

The me thod of s olut i on t o b e p r op o s e d he re i s o r iginal in

that i t i s the only one whi ch allows for c on s i de rat i on of the non

line ari ty due t o chemi c al kine t i c s . O the r approache s have e i the r

t aken an empi ri c al value of the s p re ad ing rate as known and c a lcu

late d t empe rature fi e ld s o r have de t e rmine d the s pre ad ing rate

as a func t i on of s ome heuri s t i c parame t e r whi ch i s no t re ad i ly

re late d t o o the r fundame ntal pr ope rite s(q )

. One exc ep t i on i s the

s tudy by Tari fa , e t a l ( 5 ) but the re rad iat i on i s the only me chani sm

by whi ch ene rgy i s all owe d t o be t rans fe rre d ahead of the flame .

2 0 8

The original i ty of the me thod c aus e d the author t o be c aut i ous and

t o at t emp t , in the fi rs t ins t ance , t o s olve the c as e of s urfac e

re ac t i ons rathe r than the more phy s i c ally int e re s t ing c as e of gas

phas e reac t i on s .

The frame of re fe renc e wi ll be fixed t o the mov ing flame

fr ont s o that a s te ady - s t at e pr ob lem is ob t aine d . The amb i ent

pre s s ure is uni form and no pre s sure grad ient s exi s t th roughout the

heat - up and re ac t i on z one s at the low Mach numbe r s involve d . Due

t o the t empe rature inc re as e s and dens i ty de c re a s e s as s oc i ate d wi th

the re ac t i on , the s t ream line s wi ll dive rge s omewhat . Howeve r , thi s

e f fe c t is negle c t e d in the gove rning e quat i ons and the Oseen

appr oximat i on is made wi th the c onve c t ive te rm s . Rad i at i on i s

negle c t e d in thi s mode l s inc e i t is not expe c t e d t o b e import ant

for small- s c ale f i re s , at le as t . The flow in the flame - fr ont

regi on i s c on s i de re d t o be lam inar . Al s o , the Prandt l numb e r i s

as sumed t o be ne gl igib le c ompare d t o uni ty s o that the vi s c ous

laye r i s much thinne r than the the rmal laye r . Then the momentum

e quat i on may be c ons ide re d t rivial and the gas phas e e quat i ons may

b e wri t ten as

Spe c i e saYO

E ne rgy

8TpVC

p a;V ° ( AVT )

whe re p dens i ty , c i s the s pe c i fic heat whi c hP

as s ume d i dent i c al f or all sp e c i e s , V is the ve l oc i ty of the ai r

re lat ive t o the flame fr ont , YOi s the mas s frac t i on fo r the

oxi d i z e r , T i s the tempe r ature , D i s the ma s s di ffus iv i ty , A is

the the rmal c onduc t ivi ty , x is the c oord inat e paralle l t o the

d i re c t i on of f lame pr opagat i on and y i s the c oord inate normal t o

the f lame p r opagat i on d i re c t i on .

In the s ol i d phas e , the ene rgy e quat i on i s wr i t t en as

8TS

psVFCs 5x

whe re the s ub s c r ipt s imp li e s s oli d pha s e . VF

i s the flame

s pre ad ing rate re lat ive t o the s o l i d . Not e that in the ab s ence of

natural or fo rc e d c onve c t i on V VF

' y O is t aken as the s oli d

gas s urfac e and the rat i o of s ur fac e reg re s s i on rate t o s preading

rat e is negle c te d . AS

c ons t ant may be as s ume d .

Ce r t ain mat ching c ond i t i on s wi ll b e app lie d at the s ol i d - ga s

inte rfac e . F i r s t of all , the c omb ine d f lux of oxi d i z e r due t o

d i ffus i on and c onve c t i on at the int e rfac e mus t b alance the oxi da

t i on rate at the s urfac e ; i . e . ,

l

whe re mi s the mas s f lux em i t t e d fr om the s urfac e and v is the

s t oi chi ome t ri c mas s rat i o of fue l- t o - oxi d i z e r . The gas i f i c at i on

rat e i s give n by a kine t i c l aw whi ch as sume s a fi rs t orde r de pen

dence upon lo c al oxi di z e r c onc ent rat i on and an Ar rhenius de pendenc e

upon tempe rature .

https://www.forgottenbooks.com/join

In the ab ove e quat i ons , uni t ary Lewi s numb e r i s as sume d ; howeve r ,

that as s umpt i on c an be re laxe d wi th s ome inc re as e of c omp lexi ty in

the c alculat i ons .

The foll owing nond imens i onal vari ab le s are employed

0 2

s ( 13 ) Vt x/Ll

c n pO Dc dy 1 dyZ :

a“

a—

Ll l

e T/T

whe re L K/pooc

i s a charac t e ri s t i c the rmal length . The s e lead

t o the e quat i ons gove rning the gas phas e

2as a a

as“

5?+7 ( 9 )

oYO

ogr o

gr

‘

6 5‘

8 22

ds2 ( 1 0 )

Fo r the s oli d phas e , we de fine the nond imens i onal variab le s

and parame t e r

Thi s re s ult s in the fo llowing form of the ene rgy e quat i on for the

s ol i d phas e

The b oundary c ond i t i ons at infini ty are that

8 YO

—v l as z —v oo or s —v - oo

as

—v l a s yS

—v oo or s —v - oo ( 1 2 )

The b oundary c ond i t i on s and ( 6 ) may b e c omb ine d and

t rans fo rme d t o

pro A

Yo

- eC/e- 1

is?“ tv)

‘

77e

59+ (

A)YO

e

- GC/e- l

Q,( l

clS) ( 6 l6 77: V

“

e"

c;

The sy s tem of e quat i ons and ( 1 1 ) t oge the r wi th the

boundary c ondi t i ons may be s olve d wi th the ai d of the Gre en 's

func t i on whi ch i s deve l ope d in the Appendix . A sy s tem o f nonlinea r

inte g ral e quat i ons wi th one inde pendent vari ab le are ob t aine d as

fol lows

s*

S “

9 / ( e- 1 )8 ( s ) 1 + A —QL - ( e

C

CD S

row} Is — t l) udt

whe re the de finit i ons have b e en made that

and s*

i s the pos i t i on b eyond whi ch the s ur fac e oxi d at i on no longe r

oc cur s . Fo r an infini t e ly thi ck fue l- b e d , s*

— » cp

Re ali z e that the fue l i s gas i fy ing s o that the s ur fac e is

reg re s s ing as the flame move s along i t . When the fue l - be d i s ve ry

thi c k , the po s i t i on of fue l - b e d burn — out i s s o far d owns t re am of

the flame fr ont that the exac t po s i t i on of burn- out or the exac t

thi ckne s s of the b e d i s not impor t ant . That is , ab ove a c e rt ain

fue l— b e d thi ckne s s , the s pre ad ing rat e and the fie ld s olut i on in

the f lame fr ont re gi on do not de pend up on the fue l - b e d thi ckne s s T .

In thi s range of T , the re sult s are inde pendent of s*

. Fo r a thin

fue l - be d , on the othe r hand , our e quat i ons are ac c urate i f we

c ons ide r the foll owing s i tuat i on . A s sume a fue l , of thi ckne s s T ,

i s c oate d upon an ine rt s ub s t rat e of thi ckne s s Furth erm ore ,

T 6*

s o that the t ot al thi ckne s s T +~ 6

*

x b*

. Al s o , the s ub

s t rate and the fue l have i dent i c al the rmal d i ffus iv i t i e s .

The s pre ad ing ve loc i ty V and the re fore V ) i s s t i ll unknownF

and mus t be c ons ide re d as an e igenvalue . Thi s imp li e s that A i s

an e i genvalue of the p rob lem . S inc e and ( 1 7 ) form a

sy s t em of thre e e quat i on s f or the t empe rature 8 , oxi d i z e r c on

c ent rat i onYO , and he at flux u at the s ur fac e . Howeve r , s inc e A

2 1 4

https://www.forgottenbooks.com/join

( 1 ) the re is only one inde pendent vari ab le ins te ad of two vari ab le s

and ( 2 ) the nonl inear sy s t em i s re adi ly s o lve d by the me thod of

s uc c e s s ive s ub s t i tut i ons . I f , in addi t i on t o the s pre ad ing rate

and the sur fac e value s , one wi s he s t o de t e rmine the s olut i on

through the gas and s ol i d fi e ld s , they may b e c ons t ruc t e d f rom the

s urface value s thr ough the us e of E quat i on ( A - 4 ) in the Append ix .

The sy s t em of thre e int eg ral e quat i ons may be re duc e d unde r

spe c i al c i r cum s t anc e s . In one sp e c i al c a s e , al l of the ene rgy i s

c onduc te d through the gas phas e and none i s c onduc t e d through the

s oli d phas e . In thi s c as e the he at flux in the s oli d phas e ne e d

not be de t e rmine d in a c oup le d fashi on . The re f ore , E quat i ons

( 1 6 ) and ( 1 8 ) are s olve d t oge the r ne gle c t ing E quat i on ( 1 7 ) and

s e t t ing u 0 in E quat i on

Thi s l im i t c ould b e ob t aine d by le t t ing 6 - 0 in whi ch c as e

the ke rne l in E quat i on ( 1 7 ) be c ome s infini t e y i e lding the s olut i on

u 0 . Phy s i c ally , thi s imp li e s that as the fue l - b e d be c ome s ve ry

thin no ene rgy i s c onduc te d thr ough i t . In thi s lim i t , we are le ft

wi th the fol lowing e quat i on

whi ch i s the lim i t ing form of ( 1 5 ) and mus t b e s olve d t oge the r wi th

E quat i ons ( 1 6 ) and ( 1 8 ) for 8 , Y and A .

O }

A furthe r s pe c i al sub c as e oc c ur s when c

pcS

and v 1 wi th

u 0 . Then E quat i ons ( 1 6 ) and ( 1 9 ) y ie ld

s -g Y - 9 - 1 )Yo

Yo

A eT

KOIs ' e

9

0 e Cde N —

O°

I7;( 2 0 )

h

G )

whi ch may be sub s t i tute d int o the inte g rals in ( 1 8 ) and ( 2 0 ) to

ob t ain

s*

s - 8C/ ( 8 - l )

9 x “ AO(LS -

751 )

e( q — 9 ) de

s Y ( h /vQ ) ( l - 9 ) - lo - 9 9 - 1

A a B 00

6e

whe re the de fini t i on ha s b e en made that

Yv

q 10

Q0 0

He re ( 2 2 ) and ( 23 ) may be s o lve d t oge the r for 8 and A . Afte rward ,

Y may be de t e rm ine d fr om0

Nume ri c al Me thod s

The non linear inte gral e quat i ons are s o lve d by the me thod o f

suc c e s s ive sub s t i tut i ons . A gue s s i s made at the s oluti on and

s ubs t i tute d int o the int eg rals on the r ight - hand s i de s of the e qua

t i ons . The le ft - hand s id e s of the e quat i ons as c alc ulated be c ome

the next gue s s and are sub s t i tute d int o the integ ral s fo r the next

s tep in the i te rat ive proc e s s . Thi s c ont inue s unt i l c onve rgence

O C C U F S .

7 — 1 2

Thi s te chni que has b e en suc c e s s fully empl oye d in the ab ove

ment i one d s pe c ial sub c as e whe re E quat i ons ( 2 2 ) and ( 23 ) have b een

s olve d s imult ane ous ly . The te chni que has al s o b e en employe d in

the s pe c i al c as e whe re and ( 1 9 ) are s olve d t oge the r .

A mod i f i e d form of thi s t e c hni que has b een employe d in the

s olut i on of the gene ral c as e whe re ( 1 7 ) and ( 1 8 ) are

s olve d t oge the r . Some d i ff i c ulty oc cur s be c aus e the unknown heat

flux appe ar s only in the int e grand s of ( 1 5 ) and the re for e,

the next value for u in the i t e rat i on pr oc e dure i s not imme di at e ly

c alc ulate d . Howeve r , the line ari ty of E quat i on ( 1 7 ) may be us ed

t o advantage s ince i t i s p o s s ib le t o inve rt that e quat i on and

ob t ain u as a func t i on of 8 . Thi s func t i on may then b e us e d t o

s ub s t i tute for u in The n , wi th u e liminate d , the me thod of

suc c e s s ive s ub s t i tut i ons may b e employe d . The part i cular me thod

of inve rs i on o f ( 1 7 ) involve d appr oximat ing the inte gral as a

fini t e summat i on ove r the dis c re t iz ed range of 5 . For e ach di s

c re t e value of s a di f fe rent linear algeb rai c e quat i on app li e d .

Thi s line ar algeb rai c sy s tem was inve rte d .

Re s ul t s and Di s cus s i on

C alc ul ate d re s ult s we re fi r s t ob t aine d for the s pe c ial sub

c as e whe re v 1 and u 0 . In F igure l , the s urfac e t empe rature

pr of i le de t e rmine d fr om the s o lut i on o f ( 2 2 ) and ( 23 ) i s given .

Al s o , given are the re s ult s ob t aine d fr om the s olut i on o f E quat i ons

( 1 6 ) and ( 1 8 ) wi th v . 1 I t i s s e en that as l ong as

v 1 , the re sult s o f the two me thods are in good agre ement . We

s e e that the tempe rature inc re a s e s through the flame front,

re aching a maximum , and then de c re as e s . The inc re as e oc c ur s due

https://www.forgottenbooks.com/join

As the fue l - be d thi ckne s s inc re as e s the ove rall l ength o f

the re ac t i on z one wi ll inc re as e . Howeve r , ab ove a c e r tain thi ck

ne s s ( or ab ove a c e rt ain value of B ) the t empe rature pr of i l e in

the front re gi on is e s s ent i al ly inde pendent of B . Thi s i s c le ar ly

s e en in F igure 4 , whe re the c alc ulate d tempe rature re s ult s are

s upe rimpo s e d for thre e di ffe rent fue l - be d thi ckne s s e s . The fr ont

re gi on thi ckne s s i s of the o rde r of a few the rmal lengths and

di s turb anc e s far d owns t re am d o not p ropagate up s t re am .

Nond imens i onal s p re ading rate ve r s us ene rgy re le as e i s shown

in Figure 5 and a near ly l ine ar inc re as ing de pendenc e is s e en .

Al s o s pre ad ing rate wi ll inc re as e wi th the nondimens i onal thi ckne s s

B up t o a c e r tain value of B beyond whi ch A is inde penden t o f B .

It i s al s o s e en that an inc re as e in the nondimens i onal ac t ivat i on

ene rgy re sult s in a de c re as e in the s p re ading rat e .

In F igure 6 , we s ee t empe rature re sult s f or the c as e wi th

s ol id phas e he at t rans fe r ( u O ) . The s ame charac t e r i s t i c s exi s t

as d i d for the no— s o li d- phas e - he at - t rans fe r c as e exc e p t that the

tempe rature value s are s i gni fi c ant ly re duce d by the c ooling e ffe c t

of the s ol i d . For the value s of a and a s given ,

the main e ffe c t of the s oli d phas e i s t o c o ol the re ac t i on z one

the reby de c re a s ing the s p re ading rate ; A he re ve r s us A

wi th no s o li d phas e he at t rans fe r . Re ali z e , of c our s e , that wi th

lowe r value s of a ( hight s ol i d the rmal d i ffus ivi ty or gre ate r

value s of V/VS) the s ol i d phas e wi ll b egin t o p lay a mo re import ant

r ole in t rans fe rring ene rgy ahe ad of the flame whi ch would t end t o

enhance the s pread ing rate .

The re s ult s shown in F igure 7 would ind i c at e at fi r s t s ight

that s o li d phas e he at t rans fe r make s only s light d i ffe renc e s in

the re sult s fo r oxid i z e r mas s frac t i on . The di f fe renc e would b e

made through the t empe rature whi ch mod i fi e s the re ac t i on rate .

Reali z e that again the inde pendent vari ab le s i s b as ed on a

charac t e ri s t i c the rmal lengt h whi ch de pend s upon the s pre ading

rate . The re f ore , s inc e the s pre ading rate de pend s s igni fi c ant ly

upon s oli d phas e heat t rans fe r even the oxi d i z e r mas s frac t i ons

are affe c te d . Thi s di s c us s i on als o re lat e s t o Figure 3 .

In Figure 8 , we s e e the s ur fac e he at flux pl ot t e d ve r s us

po s i t i on al ong the s ur face . The he at flux ha s i t s maximum in the

flame fr ont re gi on wi th anothe r loc al maximum oc curr ing j us t b e fore

the burnout po s i t i on s* Thi s las t maximum c oinc i de s with

regi on of surface tempe rature de c re as e a s would be expe c te d .

The c alculat i on s are p re lim inary in the s ens e that no ext en

s ive parame te r survey ha s ye t be en pe rforme d . Howeve r , i t i s fe lt

that the fe as ib i li ty o f us ing thi s int e gral te c hni que for such non

line ar c alc ulat i on s ha s be en demon s t rate d .

The autho r wi she s t o acknowle dge the s uppo rt of the Nat iona l

Ae ronauti c s and Spac e Admini s t rati on and the Nati ona l Sc ience

Foundat i on fo r the i r s upp o rt of thi s e ffort unde r C ont ra c t

NAS2 - 67 05 and G rant 0 1 3 255 4xl .

Re fe renc e s

S i r ignano , W . A . ,

”A C r i t i c al Di s c us s i on of The ori e s of Flame

Sp re ad ac r o s s S ol i d and Li qui d Fue l s " , C ombus t i on S c i ence and

Las t rina , F . A . , Mage e , R . S . and McAlevy , R . F . , Flame Spre ad

Ove r Fue l Be d s : Sol i d Phas e E ne rgy C ons ide rat i ons " ,

C ombus t i on Ins t i tut e , Pi tt sburgh , p . 93 5 .

Fe rnande z - Pe ll o , A ., Kinde lan , M . , and Wi ll i am s , F . A . ,

" Sur face Tempe rature Hi s t or i e s During Downward Propagat i on of

Flame s on PMMA She e t s " , pre p rinte d f or 1 973 Sp r ing Me e t ing of

We s te rn S t ate s Se c tion/The C ombus t i on Ins t i tut e , Apr i l 1 6 - 1 7 ,

1 973

de Ri s , J . ,

" The Sp re ad of a Laminar Diffus i on Flame

C ombus t i on Ins t i tute , Pi t t sburgh , p . 24 1 .

Tari fa , O . S . , de l No tar i o , P . P . , and Tor ra lb o , A .M . , On the

Proc e s s o f Flame Sp re ad ing Ove r the Sur fac e of Plas t i c Fue l s

on C ombus t i on , C ombus t i on Ins t i tute , Pi tt sb urgh , p . 2 2 9 .

https://www.forgottenbooks.com/join

Then we have fr om ( A- 3 )

u ( 6 , n)

Now i t only remains t o de t e rmine the G whi ch s at i s fi e s the

ab ove thre e c ondi t i on s .

C ons i de r the ad j oint e quat i on ( A- 2 ) and le t

u ( x , y ) E

Then

Av + kvX

- %Eu ] 0

s o that

C ons i de ring the cy lindr i c ally symme t ri c s olut i on for u we would

ob t ain

whi ch is a mod i fi e d Be s s e l 's e quat i on of z e ro orde r . One s olut i on

i s the modi fie d Be s s e l func t i on of z e ro orde r and s e c ond kind

krKO(—é_) whe re

lKO( Z) " _

glog §

z + y

Note that y i s E ule r 's c ons t ant

2 24

Asympt ot i c al ly we have

Ko( z ) m log z a s z 0

WKO( Z) N (E ) e as z —* CD

Now v KO(§ 17 ) is a fundamental s olut i on t o the

ad j oint e quat i on s o that c ond i t i on ( 1 ) is s at i s fi e d . It remains

t o s at i s fy c ondi t i ons ( 2 ) and C ondi t i on ( 3 ) is s at i s fi e d by

the symme t ri c re fle c t i on ab out the line y 0 . De fining

v

whi ch s at i s fi e s the b oundary c ondi t i on that -

537( x , 0 ) O as we l l as

s at i s fying the par t i al d i ffe rent i al e quat i on ( A

C ond it ion ( 2 ) i s s at i s fi ed by mult i p lying by the c ons t ant

1s o that f inally

v5

1

1}

-r ) + KO(§ r ' (A-S )

In parti c ular

l - tu ( A- 6 )Tr

The d i ffe renc e be tween up s t ream and down s t ream influence

be re ad i ly s een . C ons ide r the regi on whe re Ix- fi l ly l and

Iq l. Then

7 — 2 0

i x - gl)

l emm a - eh Tr pe e

- (w ),

n.

s o that i f x 2, we have

G N

1)l/Q

e

- k IX- fi lW K | X

and for x a, we have

1

The s olut i on at the po s i t i on x has s ome influenc e on the s olu

t i on at the p oint i and the Gre en ‘ s func t i on G i s a meas ure of that

influenc e . When 2 i s up s t re am of x ( or x the inf luenc e i s

re lat ive ly weak s inc e the re is an exponent i al de c ay as | x- fi l in

c re as e s . Howeve r , when 6 i s d owns t re am of x ( o r x the

influenc e i s s omewhat s t ronge r s ince the de c ay goe s as lx - fi l- l/g

.

On ac c ount of the c onve c t i on , a di s turb anc e t o the fie ld at the

p o s i t i on x would be fe l t mo re s t r ongly in the d own s t re am di re c t i on

than in the up s t re am di re c t i on .

Supp o s e we we re c on s i de r ing a s ol i d fue l wi th a thi ckne s s 6

that i s not ve ry much large r than a charac te r i s t i c the rmal thi ck

ne s s . Then the b oundary c ond i t i on g iven by ( 1 2 ) i s modi fi e d s o

asthat

B—E- z O at 6 6

*

/L . In the nome nc lature of thi s appendix ,

ys

we mus t impo s e a fourth c ond i t i on on our Gre en 's func t i on ; name ly

2; 0 . By the me tho d of image s the Gre en 's func t i on i s

found t o b e

https://www.forgottenbooks.com/join

Figure 1

Figure

Figure 3

https://www.forgottenbooks.com/join

E nergy release,0

Figure 5

F igure 7

23 4

https://www.forgottenbooks.com/join

in a Spark— Igni t i on E ngine

W . A . S i ri gnano

Guggenhe im Lab orator i e sPrinc e t on Unive rs i ty

Ab s t rac tA the ory b as e d upon a c onc e p t of turbulent flame p r opagat i on

has deve lope d and ha s re s ult e d in the c alc ulat i on of p re s sureve r s us c rank angle and temp e rature ve r sus b o th c rank angle andc hamb e r p o s i t i on as a func t i on of vari ous de s i gn parame te r s .

Ul t imate ly , the the ory would re s ult in the c alculat i on of NO c onc ent rat i on . Turbulent mixing oc cur s t o a s igni fi c ant ext entthroughout the chamb e r e s pe c i ally fo r large r turbulent e ddy s i z e s .

Afte r burning i s c omp le te d , the m ixing t end s t o uni formi z e thetempe rature d i s t r ibut i on s omewhat .

The c alculat i on of the c onc ent rat i on of emi s s i on s of a spark

igni t i on ( Ot t o ) engine re qui re s a knowle dge of the pre s sure and

tempe rature dependence up on spac e and t ime in the c ombus t i on

c hamb e r . The re a s on for thi s i s that s uch s pe c i e s , s uch as NO , a re

forme d in a none qui lib rium manne r and the i r exhaus t c onc ent rat i ons

c anno t b e c alc ulate d s ole ly from a knowle dge of the exhaus t t em

pe ra ture di s t ribut i on and pre s sure . In orde r t o de t e rmine p re s s ure

and temp e rature hi s t or ie s by mathemat i c al analy s i s , i t i s ne c e s s ary

t o unde r s t and the me c hani sm of flame p ropagat i on in such an e ng ine .

The flame p r opagat i on rat e de t e rmine s the rate of ene rgy re le as e

in the c ombus t i on cy linde r . Toge the r wi th the rat e of c ompre s s i on

( or expans i on ) by the pi s t on mot i on , the ene rgy re le as e rate

de te rmine s the p re s s ure and tempe rature variat i ons in the chamb e r

whi c h in turn gove rn the NO kine t i c s .

The inte re s t ing work of Lavoie , Heywood , and Ke ck ( 1 97 0 )

avoid s the que s t i on of the me chani sm of flame pr opagat ion .

E mperic a l pre s s ure re s ult s are employe d t o c alculate t empe rature s

and unb urnt mas s frac t i on a s func t i ons of s pac e and t ime fr om

c e rt ain the rm odynam ic c ons i de rat i ons . Thi s us e of empi r i c i sm made

a spe c i f i c s tatement ab out the me chani sm of flame pr opagat i on un

ne ce s s ary or re dundant . From the i r analy s i s , the flame p ropaga

t i on spe e d and the tempe rature vari at i on c ould b e c alc ulat e d . No

de tai l s c onc e rning the propagat i on me chani sm we re de duc e d . As a

c ons e quenc e of the i r s imp li fi c at i on , howeve r , they c ould not pre

di c t the c omple te dependenc ie s of the NO c oncent rat i ons upon par

ame t e r s whi ch would ente r through the de s c r i p t i on of the me chani sm

o f f lame pr opagat i on . Such parame t e r s inc lude the rpm,mixture

rat i o , s park advanc e , c ompre s s i on rat i o,

and d i s plac ement s inc e

the s e parame te r s affe c t the pre s s ure pr ofi le whi ch is taken empir

ic a l ly in that work .

I t would be mo s t us e ful , the re fo re , t o have an analy s i s whi c h

mode l s the flame pr opagat i on me chani sm and c an p re d i c t the c omple t e

de pendenc ie s of the NO c onc ent ra t i on upon the s e c ri t i c al parame

te r s . Thi s pape r d i s c us s e s s uch a mode l . It is argued in thi s

mode l that the flame propagat i on invo lve s , in an e s s ent i al manne r ,

the turbulent t rans fe r of he at ahe ad of the flame . A c alc ulati on

o f the Reynold s numb e r ( b as e d upon b ore and maximum pi s t on

Ave loc i ty ) fo r a typi c al s i tuat i on give s O ( lo whi c h j us ti fie s the

employment of a turbulent mode l . A Reynold s numbe r bas ed upon the

intake flow i s al s o high .

MODE L OF THE TURBULE NT DIFFUS IVITIE S

An analys i s of the c ombus t i on pr o c e s s in an Ot t o engine has

b e en pe rfo rme d b as e d upon the argum ent that a turbulent flame

pr opagat e s th rough the gas e ous c ombus t ib le mixture at a spe e d wh i ch

is c ont rol l ed by the rat e at whi ch the tur bulent m o t i on t rans fe r s

he at ahe ad of the flame . Thi s heat - up p ro c e s s c ont inually b ring s

the gas imme diat e ly be fore the flame t o the igni t i on point and

pr ogre s s ive c ombus t i on ( o r f lame pr opagat i on ) re sult s . The tu rbu

lent intens i ty as s o c i at e d wi th the s e e dd i e s i s re late d t o b oth the

p i s t on ve l oc i ty and the ve l oc i ty of the unburne d g as e s th rough the

intake valve . Obv i ous ly , the turbulent inte ns i ty inc re as e s wi th

rpm . One expe c t s , the re f ore , that f lame spe e d woul d inc re as e wi th

rpm in thi s m ode l . Thi s t rend , of c our s e , has b e en expe rimentally

de te rm ined . The length s c ale of the turbulenc e i s re lat e d t o the

cylinde r b ore and s t r oke d imens i ons and the valve opening s i z e .

I t i s a s sume d that s pat i al ly homogene ous , but t ime -varying

turbulenc e exi s t s in the c ombus t i on cy linde r . In part i c ular , an

e ddy di ffus iv i ty ( as sume d i dent i c al f o r b ot h mas s and he at t rans

fe r ) is taken t o b e the sum of two d i f fus ivi t i e s , one due t o

p i s t on- m ot i on- gene rate d turbulenc e and the othe r due t o int ake

f low- gene rate d turbulenc e . By d imens i onal analy s i s , i t c an be

c onc lude d that each d i ffus ivi ty i s the p roduc t of a charac t e ri s t i c

length and a charac t e r i s t i c ve loc i ty . With one di ffus ivi ty , the

charac t e r i s t i c ve l oc i ty e qual s the ( ab s olute value o f ) p i s t on

ve loc i ty at e ach ins t ant , and , wi th the s e c ond di ffus ivi ty , the

cha ra c t e r sit ic ve loc i ty i s p rop or t i onal t o the ave rage int ake gas

23 8

https://www.forgottenbooks.com/join

but t ime - varying p re s s ure . I t i s c onveni ent in the fi r s t analy s i s

t o c ons ide r a one - d imen s i onal un s t e ady p rob lem . Figure 1 ind i c at e s

two int e re s t ing mode l s f or f lame pr opagat i on . On the le ft - hand

s id e of the figure we s e e a p lanar wave p r opagat ing from the

cy l inde r he ad ( whe re the spark igni t i on oc c urs ) t oward s the pi s t on .

That i s , vari at i ons ac ro s s the c i r c ular c r o s s — s e c t i on of the

cyl inde r are neg le c te d in c ompar i s on t o vari at i ons in the axi al

d i re c t i on . Thi s i s s t ri c t ly vali d only in the c as e whe re the b ore

t o- s t roke rat i o i s much le s s than uni ty and the p rimary di re c t i on

of flame prop agat i on i s the axi al d i re c t i on . Al though , thi s is

no t re ali s t i c wi th re gard t o prac t i c al aut omot ive de s ign , the

e s s ent i al phy s i c s of the prob lem remain int ac t unde r thi s ide al i z a

t i on and b as i c t rend s s hould b e not ewo rthy . On the r ight —hand s i de ,

we mode l a cyl ind r i c al f lame p r opagat ing from the c ente r of the

chamb e r ( whe re igni t i on o c cur s ) t oward s the cy l inde r wal l s . He re

grad ient s in the ve rt i c al d i re c t i on are ne gle c t e d . Thi s i s p e rhap s

a s omewhat more re ali s t i c mode l than the p lanar flame c a s e but

s light ly more c omp lex mathemat i c ally . The p lanar m ode l was chos en

a s the fi r s t m ode l in what i s hope d t o b e c ome an impr ov ing

s uc c e s s i on of mode ls . The p rimary intent he re i s t o show the

fe as ib i li ty of c alc ulat ing the fie ld p r ope r t i e s in a c omb us t i on

chamb e r wi th turbulent flame pr opagat i on . Onc e thi s fe as ib i li ty

i s dem ons t rate d m ore re ali s t i c and more c omp lex mode l s may be

s tud i e d . In future analy s e s , one c ould t re at the two or thre e

d imens i onal pr ob lem whe re the b o re — t o - s t r oke rat i o c ould b e a mo re

re ali s t i c value .

The c ont inui ty e quat i on i s g iven as

24 0

<2 )

and the ene rgy e quat i on is g iven as

a a :l d 1 B

whe re p is the dens i ty , u is the ga s ve loc i ty , x is the axi al

dimens i on , p is the pre s sure , h is the enthalpy , T i s the t empe ra

ture , cp

i s the s pe c i fi c he at at c ons t ant p re s s ure , and Q i s the

ene rgy re leas e d pe r uni t t ime by the c ombus t i on pr oc e s s .

Afte r the intake valve c lo s e s , the amount of mas s m in the

cy linde r is fixe d .

*

The fixe d mas s , moving b oundary pr ob lem is

mo s t c onvenient ly hand le d in a Lagrangian frame of re fe renc e . The

t rans format i on

"6 II pdx

'

is made whe re x O and w O are at the cy linde r he ad whi le w

t ot al ga s mas s d ivide d by cy linde r c r os s - s e c t i onal are a oc c ur s at

the pi s t on fac e . Thi s t rans format i on e s s ent i ally re plac e s the

c ont inui ty e quat i on and le ad s t o the foll owing form of the ene rgy

e quat i on

a l d a 2 a Q( 4 ) R

T “ p = ( p ub— N t

pcpfl aw w C

p

whe re c

phas b een c ons id e re d as c ons tant . Furthe rmo re , a s suming

the pe rfe c t ga s re lat i on shi p and reali z ing a and p are func t ions

Thi s mas s m i s gene rally a we ak func t ion of rpm ac c ord ing to

empi r i c al re s ul t s given by Li chty

24 1

8 - 7

o f t ime only , we c an rewr i t e E q . ( 4 ) as

2

( 5 ) 2303 17

1tug

whe re y i s the rat i o of the s pe c i fi c heat s and R i s the gas

c ons t ant .

The fi r s t t e rm on the ri ght - hand s i d e of E q . ( 5 ) repre s ent s

the t urbulent t rans fe r of ene rgy whi le the s e c ond t e rm re pre s ent s

the ene rgy re le as e . I f b o th o f the s e t e rm s we r e z e ro , no ent r opy

i s pr oduc e d and the t empe rature and pre s s ure foll ow the i s ent r op i c

re lat i ons hip . ( Inde e d , i t i s s e en that s e t t ing the le ft — hand s i de

of the e quat i on t o z e ro and int eg rat ing y i e ld s that pl - y/y T i s a

c ons t ant . )

The b oundary c ondi t i on s on E q . ( 5 ) are that ne glig ib le

ene rgy i s t ran s fe rre d through the p i s t on fac e and cy l inde r he ad ,

name ly

( 6 ) g; ( t , o) o

and

( 7 ) 0

Furthe rmore,

the init i al tempe rature d i s t r ibut i on i s s pe c i f i e d ;

s ome t emp e rature exi s t s in the ne ighb orhood of the s p ark p lug j us t

afte r igni t i on w ith l owe r tempe rature s away from the s park p lug .

The p re s sure c an b e re lat e d t o the inte g ral of the t empe ra

ture d i s t r ibut i on . In part i c ular , f rom the pe r fe c t gas l aw we have

p =pRT

= % RT

o r , integ rat ing ove r the t ot al gas v olume , we ob t ain

https://www.forgottenbooks.com/join

( 8 a ) Pv

Of c our se , now the ini t i al c ond i t i on on E q . ( 5 a ) amount s t o the

s pe c i fi c at i on of K j us t fo ll owing igni t i on .

Not e that E q . ( 5 a ) now ha s the e ffe c t of the rate of change

of p re s sure c onc e ale d wi thin the de fini t i on of K . The vari at i on

of K from uni ty oc c ur s due t o the nonis ent ropic p roc e s s e s of turbu

lent d i f fus i on of ene rgy and chemi c al ene rgy re le as e .

I t is re ali z e d that the ene rgy re le a s e Q de pends up on b oth

the tempe rature and c onc ent rat i on . As sum ing a s e c ond orde r re

-X

ac t i on , we c an show that wi th a s t o i chi ome t r i c mixture

( 9 ) Q a peee

' E /RT

whe re E i s the ac t ivat i on ene rgy , 5 i s the mas s frac t i on of unburnt

gas e s , and a is a pre - exp onent i al c ons t ant ( whi ch c ould b e as s ume d

dep endent up on tempe rature i f de s i re d ) . Of c our s e , i t i s p os s ib le

t o u se s ome o the r re lat i on sh ip for Q ins t e ad of E q . ( 9 ) i f s o

de s i re d .

S PE C IE S E QUATI ON

I t i s c le ar f rom E q . ( 5 a ) and E q . ( 9 ) that the mas s frac t i on

of unburnt s pe c ie s mus t be de te rmine d a s a func t i on of spac e and

t ime . The gove rning e quat i on i s

E quat i on ( 9 ) c ould b e e as i ly m odi fi e d t o ac c ount fo r offs toi chi ome t ri c c as e s o r c ould b e re plac e d by a sy s tem of e quat i onst o de s c r ib e de t ai le d k ine t i c s .

8 - 1 0

( 1 0 )(

95; 53

7 8201 25)

whe re Q is the chemi c al ene rgy pe r uni t mas s of unburnt ga s . The

fi r s t te rm on the right -hand s ide repr e s ent s the turbulent diffu

s i on of the unburnt s pe c i e s whi le the s e c ond te rm repre s ent s the

dep le t i on o f the s pe c i e s due t o c ombus t i on . Note that the turbu

lent d i ffus ivi ty for mas s t rans fe r ha s b e en as sume d e qual t o the

d i ffus ivi ty for ene rgy t rans fe r .

Us ing the pe rfe c t ga s law and t rans f orm ing t o the non- dimen

siona l variab le s , E q . ( 1 0 ) may be rewr i t t en a s

i)2 as

]Q.

( 1 1 )3 8

3: P BR PR mB E0

The b oundary c ondit i ons are

( 1 2 )

and

( 13 )as

whi ch imply that no mas s d i f fus e s through the cy linde r he ad or

pi s t on fac e . The ini t i al c ond i t i on i s given as e 1 eve rywhe re

thr oughout the c ombus t i on chamb e r when the value of C i s co

exc ep t

fo r a small regi on ne ar the s park p lug whe re igni t i on oc cur s .

Rathe r than int egrat ing E q . ( 5a ) and E q . ( 1 1 ) s imul taneous ly ,

i t i s c onvenient t o de fine

( 1 4 ) e = x +Q

8cpTo

Then c omb inat i on of E q s . ( 5 a ) , ( 6 a ) , and ( 7 a ) wi th E q s .

and ( 1 3 ) le ad s to the foll owing e quat i on

86 A a 2/ 8 A 88 Q l( 1 5 ) PE

. (v3) E

' P y‘

gfi n'

gfilp

T [ P Y/y l ]

w i th the b oundary c ond it i ons

( 1 6 ) gg( c o ) o

and

86( 1 7 ) B

‘

fi( C2 1 ) O

The ini t i al c ond i t i on fo r B i s re adi ly de t e rmine d g iven the ini t i al

c ondi t i ons for s and K .

In a c ons t ant p re s s ure pr oc e s s i t would foll ow that the E q .

( 1 5 ) and the b oundary c ondi t i ons E q s . ( 1 6 ) and ( 1 7 ) are s at i s fi e d

by the s olut i on 8 c ons t ant . Wi th vary ing pre s sure , howeve r , s ome

var iat i on in 8 o c c ur s . The vari at i on in 8 i s s ub s t ant i al ly le s s

than the vari at i on in s and , fo r thi s rea s on , nume ri c al e r ror s ar e

m inim i z e d when E q s . and ( 1 7 ) are emp loye d in l i eu of

E q s . and The re fo re , the sy s t em of part i al

di ffe rent i al e quat i ons whi c h are t o b e s olve d nume ri c ally are E q s .

( 5 a ) and ( 1 5 ) s ub j e c t t o the de f ini t i ons E q s . ( 9 ) and t o the

b oundary c ondi t i ons E q s . ( 6 a ) , ( 7 a ) , and and t o the

app r opr i at e ini t i al c ondi t i on s .

NUME RICAL INTE GRATION OF THE E QUATIONS

S inc e the c oe f fi c i ent s of the s e c ond de r ivat ive te rm s in

E q s . ( 5 a ) and ( 1 5 ) are s t r ong func t i ons o f t ime or c rank angle Q,

the s te p - s i z e AC re qui re d f o r ac c uracy c ould vary sub s t ant i al ly

wi th C In orde r t o p r oc e e d wi th c ons tant s t e p - s i z e , i t i s c on

venient t o make a c e r tain t rans f ormat i on ; i . e . ,

https://www.forgottenbooks.com/join

and the c ond it i ons

( 2 2 ) gzg ( M ) L i?( c, o ) g; (m ) o ,

( 23 ) K ( O , n) and B ( O , n) s pe c i f i e d

( 24 ) P ( l/V ) Gdn ( l/V ) Kd

E quat i ons ( 1 9 ) and ( 2 0 ) have b e en p lac e d int o a fini t e

d i ffe renc e form by me ans of the me thod of quas i - line ari zat i on

A thre e — point formula i s emp loye d for the d i ffe renc e

re pre s entat i on of the fi r s t de rivat ive s wi th re s pe c t t o z . Thi s

al lows s e c ond- orde r ac c uracy in A z . n and j are int ege r s s uc h

that we have the re lat i on s 2 na z and q jAn. 2 i s an inte ge r

denot ing the s t ep in the i te rat ive s cheme t o be de s c r ib e d . The

d i ffe renc e e quat i on s ob t aine d are

t z t+ l z z+ 1 z2 A .K B .K c .K D5 )

n : J n ) J+ l n y a n , J n : J" l

and

1 z £ + l 2 fi2 6 F

fl z+. G H K

n , JBn , J+ l n , J

Bn , J n , J

Bn , J— l n , J

whe re the de fini t i ons have b e en made that

n , J

z 2 3 z[ ( B

H , JKn, 3

) " 2 ( Bn, a

2(“S

n/2AZ

No t e that when n l the las t t e rm s in Dz and K

g are re plac e dn , J n , J

byKO and s

oAl s o when n 1 the las t t e rm in

1

e ach of B fi J and Gfi j i s re p lac e d by The b oundary c ondi

t i ons E q . ( 2 2 ) are rep lac e d by

2+ l £ + l 2+ l a+ iKn ,

- lKn , l

Kn , j+ l

Kn , j- l

£+ l £+ l £+ lBn ,

-l6n , 1

Bn , j+ l

Bn , j- l

with the ini t i al c ondi t i onsKO 3

and Bo 3

g iven for all j o f

inte re s t . E quat i on ( 24 ) i s r e pre s ente d by

( 2 8 ) PZ+ 1

( l/V )A (l( K3

K0

K3

n nT] 2 n , o n , j n , j

The me tho d of quas i - lineari z at i on i s a t e chni que fo r im

p r ov ing the ac c uracy of the c oe ffi c ient s in E q s . ( 2 5 ) and ( 2 6 ) in

an it e rat ive manne r . At e ach value of n , the s olut i ons fo r all j

are ob t aine d in e ach s t ep of the i t e rat i on unt i l s at i s fac t o ry c on

ve rgence oc cur s . Then , the s olut i on for the next value of n i s

de t e rmine d . The i te rat i on b eg ins by cho o s ing )g+ 1

n , J

whi ch i s the final value from the p revi ous i t e rat i on . The

I'

l— l , ,j

c oe ffi c i ent s in E q s . ( 2 5 ) and ( 2 6 ) have be en de fine d s uch that ,

2 5 0

https://www.forgottenbooks.com/join

The param e te r s in the b as i c c as e ( C as e I ) we re chos en a s

f ol lows

9c

1 x 1 05

- 2 00

X

y= 1 3

Qc Tp o

No te that A i s d i re c t ly p rop or t i onal t o the p re - exponenti al

c ons t ant in the c hemi c al kine t i c law and inve r s e ly p rop or t i onal t o

the rpm value . The given value of A re pre s ent s a c ons t ant of the

o rde r of 1 01 3

cmB/ (mole — s e c ) and an rpm value of 2 0 0 0 . The value

of 9c

imp l i e s a value of the ac t ivat i on ene rgy whi ch is ab out 2 0

kc al

The re s ult s fo r the unburnt mas s f rac t i on ve r s us chambe r

po s i t i on f o r vari ous c rank angle s are p re s ent e d in Figure 2 . They

ind i c ate that burning begins near the s park p lug and p ropagat e s

t oward s the pi s t on . The flame has a c e r t ain thi ckne s s and re ac t i on

and turbulent mas s d i ffus i on are s i gni fi c ant thr oughout s ome

por t i on of the chambe r at e ach ins t ant .

In Figure 3 , K ve r sus chambe r p o s i t i on i s p lot te d ve r s us

vari ous c rank angle s . Again , the pr opagat i on of a flame s t ruc ture

is ind i c ate d . A t the end of burning , K ha s a ne ar ly uni fo rm value

b e tween a . o and due t o m ixing e ffe c t s . Thi s impli e s that

t empe rature gradient s t end t o b e e liminat e d by the mixing . E ffe c t s

of wall quenching and he at t rans fe r are not inc lude d in thi s

2 5 2

analy s i s , howeve r .

value of K af te r the c omple t i on of burning at C

an i s ent ropi c expans i on is oc c urring .

The ne arly uni form and ne ar ly t ime - inde pendent

00

implie s that

Tempe rature ve r s us chambe r po s i t ion for vari ous c rank ang le s

is p lo t t e d in Figure 4 . The p revi ous ly - menti one d t rend s are al s o

dem on s t rate d the re but i t i s al s o indi c at e d that aft e r c ombus t i on

is c omple te d the t empe rature de c re as e s uni formly during the

expans i on proc e s s .

Now ,

vari ous parame te r s .

Tab le 1 .

a s tudy may b e pe rfo rme d of the e ffe c t s of changing

A li s t ing of the p arame te r s urvey i s given in

Tab le 1 Summary o f the Parame te r Survey

Cas e

I L 2 s x ufi

I I 1 . 2 8 x 1 05

III

IV 6 .4 0 >< 1 o5

v 1 . 2s >< 1 o5

vi

C0

— 2 0

- 2 O

- 2 0

— 2 O

- 2 O

The e f fe c t s of change s in the rat i o of s t roke - t o— e d dy s i z e

( C as e II ) are demons t rate d in Figure s 5 and 6 . I t i s s e en that an

inc re as e ( de c re as e ) in the e ddy s i z e imp lie s a de c re as e ( an

inc re as e ) in the burning angle . Thi s ind i c ate s that de s ign modifi

c at i ons whi ch c an c hange e ddy s i z e s hould have a p rofound e ffe c t

upon burning angle .

ture s would not be as profound but s i gn i fi c ant .

The e ffe c t upon pe ak pre s sure s and tempe ra

Anothe r

8 - 1 9

int e rpre t at i on is that a de c re as e in the s t r oke ( or maximum chambe r

length ) imp li e s a shor te r t rave l d i s tanc e fo r the f lame and the re

fo re a smal le r burning angle . Re al i z e that in the s e c alculat i on s ,

the c ompr e s s i on rat i o wa s he ld c ons t ant as the s t r oke wa s var i e d .

Again peak t empe rature s we re no t t oo s ens i t ive t o the parame te r

change .

Figure 7 ( C as e II I ) show s the e ffe c t of inc re as ing A t o the

value of Thi s c as e c an be viewe d e i the r as an inc re as e

in the pre - exp onent i al kine t i c c ons t ant or a de c re as e in the rpm

value . Although the re sult s indi c at e s ome de c re as e in the burning

angle , i t i s not as large as would oc cur i f the c ombus t i on pr oce s s

we re c ons t ant in durat i on . The re sult s ind i c ate that as the rpm

inc re as e s , the rat e of turbulent mixing inc re as e s due t o inc re as e s

in the pi s t on ve loc i ty and in the ve l oc i ty of the mixture during

int ake ( i f the mas s of the charge we re only we akly de pendent up on

rpm ) . The burning angle i s much le s s s ens i t ive t o the value of A

than t o the value of 9C

. Pe ak t empe rature s are not t oo s ens i t ive

t o the value of A or the value of QC

.

In Figure 8 , we s e e the e ffe c t of inc re as ing the nondimen

siona l ac t ivat i on ene rgy 9c

t o a value of and inc re as ing A t o

( C as e IV ) . Both parame te r s mus t b e c hange d s imult ane

ou s ly i f the burning angle i s t o remain at a re ali s t i c value .

Inc re as ing 9c

tend s t o s low down f lame p ropagat i on whi le inc re as ing

A re sult s in a fas t e r f lame p ropagat i on .

No te that in Figure 9 , the re s ult s for C as e V are plot t e d .

He re , the e ffe c t of de c re as ing the c ompre s s i on rat i o X t o the value

O f i s c ons i de re d . The de c re as e in the c ompre s s i on rat i o

2 54

https://www.forgottenbooks.com/join

ne ce s s ary ini t i al c ond it i ons for the intens i fi c at i on of the tu rbu

lenc e by the p i s t on mot i on .

The inte re s t ing re s ult s ob t aine d are that the final value of

K and the peak value of tempe rature 9 we re in s ens i t ive to parame t e r

change s . Re ali ze that K as de fine d b e f ore E q . ( 5 a ) c an be di re c t

ly re late d t o the ent ropy , s o that the impl i c at i on is that the

final ent ropy i s ins ens i t ive t o p arame t e r change s . Thi s i s not

surpr i s ing s inc e the c ombus t i on p roc e s s i s ve ry ne ar ly a c ons t ant

volume p roc e s s . That i s , during c ombus t i on , the p i s t on is ne ar

t op de ad c ente r and m oving s lowly s o that ve ry li t t le work i s d one

by ( or on ) the p i s t on . The re f ore , wi th the chemi c al ene rgy t o b e

re le as e d given , i t i s found that the chem i c al kine t i c c ons t ant s ,

the rpm value , e ddy - s i z e , e t c . have li t t le e f fe c t up on t empe rature

and ent r opy at the end of the c ombus t i on p r oc e s s . The pe ak

t empe rature and the tempe rature at the end o f the c ombus t i on

proc e s s we re ve ry s im i lar in the c as e s c alc ulate d he re .

At thi s po int , i t i s p o s s ib le t o us e the s e pre s s ure and

t empe rature re sult s t o c alc ulate the c onc ent rat i on of NO as a

fun c t i on of c rank angle and chambe r po s i t i on . In par t i c ular , the

c onc ent rat i on in the emi s s i on s c ould b e c alc ul ate d . The s e c alcula

t i ons are intende d for the near future .

The re i s an e arl i e r ve r s i on of thi s pape r ( S i rignano

whi ch c ome s t o s omewhat di f fe rent c onc lus i ons . In the c al cula

t i ons p re s ent ed the re , large r e ddie s and s lowe r chem i c al kine t i c s

we re employe d . Thi s re s ult e d in " thi ck" flame s . The pre s ent

2 56

c alculat i ons which re s ult in thinne r flame s are now fe lt t o b e

more re ali s t i c .

The mode l c ould b e extende d in the future by c ons i de ring the

e ffe c t s o f he at t rans fe r , de pendenc e of the s pe c i fi c he at upon

tempe rature and c onc ent rat i on , and more re ali s t i c ge ome t ri e s for

flame p rop agat i on . An int e re s t ing app li c at i on of a s imi lar mode l

of flame p ropagat i on t o the Wanke l c ombus t i on proc e s s i s d i s c us s e d

in anothe r pape r by Brac c o and S i rignano The re , in fac t ,

the c ombus t i on chambe r i s mo re re as onab ly mode lle d in a one

dimens i onal manne r than in the re c ipr oc at ing eng ine c as e .

Di s cus s i ons wi th Drs . J . Heywood and F B ra c c o on thi s poin twe re mos t us e ful .

RE FE RE NC E S

B rac c o , F .V . , and S i rignano , N . A ., The ore t i c al Analy s i s of Wanke l

Fay , J .A . , and Kaye , H"A Finit e Di ffe renc e Solut i on o f S imi l ar

Non- E qui l ib rium Boundary Laye rs " , AIAA Journal 5 , pp . 1 949

1 954 .

Lavoi e , G .A . , Heywo od , J . B ., and Ke ck , J . C E xpe riment al and

The ore t i c al S tudy of Ni t r i c Oxide F orm at i on in Inte rnal

Feb . 1 97 0 , pp . 3 1 3 - 3 2 6 .

Libby , P . A . and Chen , K . K . ,

”Rem arks on Qua s il ine a riz a tion Appl i e d

in Boundary— Laye r C alc ulat i ons " , AIAA Journal 4 , 1 966 ,

PP 93 7 - 93 9 .

McG raw - Hi ll .

Si r ignano , N . A . ,

" One - Dimens i onal Analy s i s of C ombus t i on in a

Augus t 3 - 5 , 1 97 1 .

https://www.forgottenbooks.com/join

0 5 0 6 0 7

n chamber position

Figure 2

0 1 0 3 O 5 O 6

n chamber position0 8 1 0 1 2

7?chamber positlon

Figure 4

https://www.forgottenbooks.com/join

n chamber posmon

Figure 6

0 1 0 2 0 3

n chamber posmon

0 8

OA 0 5 0 6 (1 7 (1 8

n chamber posmon

Figure 8

https://www.forgottenbooks.com/join

n chamber positlon

F igure 1 0

— 20 - 1 5 - 1O -5 0 + 5 + 1 0

C rank angle (degrees)+ 1 5 + 20 + 25 + 30

The Mas s Burning Rate of S ingle C oal Par t i c le s

I rvin Gl as sman

Depar tment of Ae r o s pac e and Me chani c al S c i enc e sPrinc e t on Unive rs i ty

Princ e t on , N . J . 0 8 54 0

Ab s t rac tThe burning rate of c oal par t i c le s are exam ine d unde r two

di f fe rent s e t s of i de al c ond it i ons ( 1 ) an as h- fre e mate ri alunde rg oing qua s i - s t eady burning in whi ch the kine t i c s of oxi dat i onon the s urfac e are fas t wi th re s pe c t t o the d i ffus i onal t ime foroxi d i z ing mate ri al t o re ach the s ur fac e and ( 2 ) an as h- fo rm ingc oal in whi c h di ffus i on through the as h c ont ro l s . S imp le m od i fi c at i on o f analy s e s alre ady in the l i t e rature s how that fo r the a sh

f re e c ondi t i on , the mas s burning rat e pe r uni t are a i s p r op or t i onalt o the mas s frac t i on of the f re e s t re am oxygen t o the fi r s t p owe rand for the inte g ral ash c ondi t i on the burning rat e i s p r opor t i onalt o the s quare r oot of oxygen mas s frac t i on . The burning rate ofan as h- free par t i c le i s al s o shown t o be a fun c t i on of the chemi c alt rans f ormat i on at the s urface . If C O form s , the burning rate i stwi c e the value that would be ob t aine d i f CO

2forme d .

1 . Int roduc t i on

The renewe d inte re s t in c oal c ombus t i on mot ivat e d thi s p ape r

whi ch is e s s ent i ally a re - analy s i s of mas s burning rate de te rm ina

t i ons fo r c e rt ain uni que pr ope rt i e s of c oal . The re are two c ond i

t i ons examine d . The fi r s t c onc e rns the burning of a sh- f re e c oal

unde r the as sumpt i on that the he t e rgene ou s oxi dat i on at the c oal

s urfac e i s fas t wi th re s pe c t t o the rate at whi c h the oxi di z ing

mate ri al is b rought t o the s urfac e . Thi s as sump t i on is val i d fo r

large par t i c le s at high tempe rature s - the m os t prac t i c al c as e o f

c oal c ombus t i on . Mul c ahy and Sm i th [ 1 ] have s hown that for pul

ve riz e d c oal even at high tempe rature s , the s ur fac e oxi dat i on

27 0

https://www.forgottenbooks.com/join

G0

i s the mas s c onsumpt i on rat e pe r uni t are a ( g/se c cm2) . k

Sis

he t e r ogene ous s pe c i fi c re ac t i on rate c ons tant whi c h inc lude s the

s urfac e are a and thus ha s uni t s of ve l oc i ty kS

( k/S )

whe re k i s the ord inary Ar rhenius rat e c ons t ant for a fi rs t orde r

re ac t i on ( s e c— l) and S i s the s urfac e are a t o volume rat i o for the

s ol id par t i c le ( cm2/c h

Dis the mas s d i ffus i on c oe ffi c ient

and a s de f ine d mus t have the uni t s of ve l oc i ty It i s

inhe rent f rom E quat i on ( 1 ) that s urfac e kine t i c s are a s s ume d fi r s t

or de r wi th re s pe c t t o the oxygen c onc e nt rat i on . CO

i s the oxygen

c onc ent rat i on The fur the r sub s c r i p t s s and 0 ) re fe r t o

the sur fac e and fre e s t re am re s pe c t ive ly . The s toi chi ome t r i c

re lat i on be tween the oxi d i z e r and fue l c an b e wr i t t en as

( 2 ) G Cf/i

whe re G i s the fue l c ons ump t ion rat e pe r uni t are a and i i s thef

mas s s t oi chi ome t ri c index . Thus E quat i on ( 1 ) may b e wr i tt en as

( 3 ) i G i ks f

j mO

i hD (

D (mO

m G, 0 3 f

in whi ch p is the t o tal gas e ous dens i ty and mo

the mas s frac t i on of

oxygen . I t i s , of c our s e , de s i rab le t o expre s s the mas s c ons ump

t i on rat e of the fue l in te rm s of the known fre e s t re am c ondi t i on ,

mo 0 3

° Thi s re sult c an be ob t aine d by s olving the two mi ddle te rm s

in E quat i on ( 3 ) for m The s imple algeb rai c re sult i s

( 4 ) m

Sub s t i tut ing E quat i on ( 4 ) int o one ob t ains

( 5 ) Gf

i p i p HID/ (1 + ( h

D/k

SH Jm

o

I t should be note d that ( hD/k

s) is a Damkohle r numb e r . When the

chem i c al rate s are fas t wi th re s pe c t t o the d i ffus i on rate , small

Damkohler numbe r ,

ks

hD

then E quat ion ( 5 ) b e c ome s

( 6 ) Gf

z i p t o, oo

or from E quat i on ( 4 )

( 7 ) m

and mo 5

may be a s sume d c lo s e t o z e r o . When the chemi c al rate s

are s low c ompare d t o the di f fus i on rat e s , large Damkohler numbe r ,

ks

hD

E quat ion ( 5 ) g ive s

( 8 ) Gf

= i ks p

m, m

and E quat i on ( 4 ) show s that

( 9 ) m mo , s

Thus for the c hemi c al rate c ont roll ing the mas s c onsump tion rate

i s found to be fi rs t ord e r wi th re s pe c t t o the f re e s t ream oxygen

mas s frac t i on and i s a d i re c t c ons e quence of the as s umpt ion of

fi rs t ord e r kine t i c s .

Inde e d , i t appe ars fr om E quat i on for di f fus i on c ont rol

l ing , that the c onsump t i on rat e is f i r s t orde r wi th re s pe c t t o the

f re e s t ream oxygen mas s fr ac t i on a s we ll . Al though thi s inde e d

wi ll turn out t o b e the c as e f or c arb on , i t i s not apparent b e c a u s

hDmus t b e more c le ar ly evaluat e d s inc e the re i s d i ffu s i on of a

gas e ous pr oduc t from the s ur fac e t o the f re e s t re am . Howeve r , for

c arb on oxi dat i on i t wi ll now b e shown that thi s flux from the

s urfac e i s small enough not t o al te r he at or mas s d i ffus i on t o or

from the s ur fac e .

In or de r t o e lab orate on thi s po int i t i s int e re s t ing t o

examine the burning rate of a volat i le fue l dr ople t in a qui e s c ent

atmos phe re as ini t i ally g iven by Spalding Spalding has shown

that

( 1 0 ) of

( Dp/r ) 1 n ( 1 + B )

whe re D i s the mole c ular di ffus i on c oe ff i c i ent r the

part i c le radius ( cm ) , and B the t rans fe r numb e r . It is c onvenient

t o wr it e B in a form fi r s t wr i t t en by Black shear [4 ] and revi ewe d

by t he author [ 5 ]

i m +~ m

( 1 1 )o a ) f s

- m H

( 1 2 >

( 1 3 ) Bof

whe re H i s the he at of c ombus t i on of the fue l ( c a l/g ) and LV

i s

the latent he at of evap orat i on The s e re s ult s evolve fr om

274

https://www.forgottenbooks.com/join

f rom the le ading e dge of the f lat p late , and i s a m od ifi ed

Blas ius func t i on whi ch i s al s o a func t i on of B . The oxygen de

pendence is in and w i ll b e di s c us s e d late r .

The s t agnant fi lm c as e , whi c h re pre s ent s c onve c t ive flow

paral le l t o the mas s evolving fr om the fue l s ur fac e , give s an

expre s s i on s imi lar t o E quat i on i . s . ,

( 1 8 ) of ( Dp/6 ) 1 n ( 1 + B )

whe re 6 i s the b oundary laye r thi ckne s s . Of c our s e , fo r the

quie s c ent c as e , as found in he at t ran s fe r , 6 r . A de fini t i on of

hD

c ould be

( 1 9 ) h E D/6D

The app r oximat i on g iven by E quat ion ( 1 8 ) i s in re ali ty qui te good

be c aus e B i s small .

To evaluate B from E quat i on ( 1 5 ) the value of i mus t b e

known . The mode l f or c arb on c ombus t i on at high t empe rature s

e lab orate d up on by C off in and B rokaw [ 7 ] has b e en gene rally

ac c e p te d . The c onc e pt i s that C O fo rm s at the s urfac e d i ffus e s

away and i s oxi d i z e d t o CO2

in the gas phas e . The C0 di ffus e s t o2

the s urface , is the e s s ent i al oxi di z ing agent and i s re duc e d t o C O

by the B oudoua rd re ac t i on

C + 2 0 0

No oxygen e s s ent i ally re ache s the sur fac e , i t is c onsume d by the

C O in the gas phas e . In thi s c as e i t has b een shown expli c i t ly

that the s t oi chi ome t ri c index is or Thi s re sult

276

may be s e en by app lying the Law of He at Summat i on to the re ac t i ons

in the ove rall sy s t em , i . e . ,

10 0

§02

CO2

0 02

C — > 2 CO

C + -

2-O2

—+ CO

If C O form s he te rogene ous ly at the surfac e then the2

s to i chi ome t ry is

10 0

502

CO2

For pure c arb on, i t is mo s t l ike ly that only C O fo rm s at the s ur

face . Howeve r , impuri t i e s c ould he te rogene ous ly c atalys t s ome CO

t o 0 02

. Thus , although i t is re c ommende d that i b e c ho sen e qual

t o i t mus t b e re ali z e d that for p rac t i c al c oal s i t s exac t

value may be s omewhat lowe r .

From e i the r of the ab ove re sult s i t i s ve ry apparent that B

i s a numbe r small c ompare d t o one for c ombus t i on wi th ai r . S inc e

mo 00

for ai r then B t o Thu s

( 2 0 ) ln ( 1 + B ) g 8

and E quat i on ( 1 8 ) be c ome s

( 2 1 ) of

( pp/6 ) B ( Up/6 ) im

whi c h wi th E quat i on ( 1 9 ) take s the fo rm

( 2 2 ) Gf= h

Dim

o , CDp

the s ame re sult as given by E quat ion The phy s i c al meaning of

277

smal l B is that the s urfac e e ff lux d oe s not have any e ffe c t on the

d i ffus i onal p roc e s s e s .

F or smal l B , E quat i on ( 1 7 ) al s o c an b e re duc e d t o the s ame

fo rm a s E quat i on B appe ar s in an analy t i c al manne r in the

b oundary c ondi t i on fo r the B las ius func t i on . The re f ore , one may

as s ume a s analy t i c in B and , furthe r , fo r the small B c a s e,

d i re c t ly p rop or t i onal t o B . Thus the mas s burning rate ha s the

s ame oxi d i z e r dependency f or the part i c le burning in a c onve c t ive

atmo sphe re as in a quie s c ent atm o sphe re .

From E quat i on one ob s e rve s that the burning rat e i s not

only di re c t ly p rop or t i onal t o the oxygen mas s f rac t i on b ut al s o

the s t o i chi ome t ri c index . C ons e quent ly whe the r CO or C02

fo rms on

the s urfac e or not i s c ruc i al b e c aus e the burning rat e change s by

a fac t or of two . One c ould have intui t ive ly p re di c te d thi s re sul t

be c aus e t o fo rm 0 02

one mus t di ffus e twi c e as much oxygen t o the

s urfac e .

The re s ult fo r the dep endency wi th re s pe c t t o oxygen ha s b e en

ob taine d by much more s ophi s t i c at e d analy s e s Many

inve s t igat or s have c ar r ie d out de t ai le d mathemat i c al analy s e s of

the ab lat i on of c arb on and of he te rogene ous ly c at aly z e d sy s tem s .

The purp o s e he re was t o show that s inc e the t rans fe r numbe r c ould

b e shown small c ompare d t o one , the s imp le analy s e s by Frank

Kamenet skii g ive the pr ope r oxi di z e r mas s frac t i on de pendency .

https://www.forgottenbooks.com/join

whe re pCis the dens i ty of the c oal . E quat ing E quat i ons

one ob tains

m

( 27 ) pC (dx/dt ) i D

Ap0

310 0

Integrat ing and s olving fo r x , one ob tains

( 2 8 ) x ( 2 DAp

zim t/pc )

0 at t E quat i on ( 2 8 ) i s c omb ine d wi th E quat i on

1 2Gf

( DAp pc

1 mo CD

/2 t )

Thus in the c as e of an a sh fo rm ing c o al , the burning rat e de

c reas e s wi th t ime and p rop or t i onal t o the oxygen mas s f rac t i on t o

the one - hal f powe r . S imi larly , i t is p r opor t i onal t o

IV . C onc lus i ons

For the c as e of non— a sh fo rm ing c oal par t i c le s burning at

high t empe rature s in e i the r a quie s c ent o r c onve c t ive atm o sphe r e ,

i t ha s b een shown in a s imple manne r that the burning rate i s

di re c t ly pr op or t i onal t o the oxygen mas s frac t i on . Fo r c oal part i

c le s whi c h form an a sh , the burning rate i s pr op or t i onal t o the

s quare root of the oxygen mas s f rac t i on .

The s ur fac e re ac t i on i s import ant in de t e rm ining the burning

rat e as we ll . If C O form s at the s urfac e in the a sh- f re e c as e , the

burning rate i s twi c e a s fa s t a s i f CO2

form s .

V . Re fe rence s

Mul

ga

hy , M . F . R . and I . W . Smi th , Rev . Pure App l . Chem . 1 9 , 8 1

( 1 9 9

Frank - Kamene t skii, D .A . ,

" Di ffus i on and He at E xchange inChemi c al Kine t i c s , Chap . I I , Princ e t on Unive rs i ty Pre s s ,

Princ e t on , N . J .

Spald ing , D . B ., Some Fundamental s of C ombus t i on , Chap . 4 ,

But te rworth , Lond on

Black she ar , P . L . , Jr ., An Int roduc t i on t o C ombus t i on

,

Chap . V , De p t . of Me c h . E ng ., Univ . of Minne s ot a

,

Minne apol i s , Minn .

G las sman , Irv in , C ombus t i on , Chap s . 6 and 9 , Ac ademicPre s s , New York

E mmons , H .W ., z . Angew . Math . Me c h . 3 6 , 6 0

Coffi

n, K . P . and R . S . Brokaw , Te ch . Note 3 9 29

1 957

Chung , P .M .,

" Chemi c ally Re ac t ing Non—E qui l ib rium BoundaryLaye r s ,

" p . 1 3 8 , in "Advanc e s in He at Trans fe r ,

" e di t ed byJ . P . Hartne t t and T. F . I rvine , Jr . , Ac adem i c Pre s s , New York

( 1 965 )

Ubhayaka r , S . K . , C ombus t i on and Flame 2 6 , 23

Knorre , G . F . , K . M . Are f 'yev , and A . G . B lokh,

"Theor

zof

95 68C ombus t i on Proc e s s e s ,

" Chap . 24 , Trans l . FTD— HT- 23by Fo re ign Te ch . Div . , Wright - Pat te r s on AFB , Ohi o ( 1 968 )

2 8 1

S tud ie s of Hyd roc arb on Oxi dat i on in a Fl ow Re ac t or*

I . Glas sman , F . L . Drye r and R . C ohen

Guggenhe im Lab o rat o rie sand

C ent e r f or E nvi r onment al S tudie sPrinc e t on Unive r sit

Princ e t on , N . J . 0 8 5 O

I . Int roduc t i on

Re c ent c onc e rns ab out ene rgy ne e d s and the as s oc i at e d envi

ronment a l pr ob lem s ha s again foc us e d at tent i on on the rathe r

s tart ling fac t that afte r burning hyd r oc arb ons f or ab out a ce ntury

a tho rough unde rs t anding of the i r high tempe rature oxi dat i on

charac te r i s t i c s s t i ll doe s not exi s t . The c ent ral thrus t of a p ro

gram a t Prince t on on the hom ogene ous ga s phas e re ac t i on kine t i c s

of hyd roc arb ons at high t empe rature s i s t o c ont ribut e t o thi s

unde r s t anding by us e o f a turbulent flow re ac t or . E arli e r work on

me thane and c arb on monoxi de oxi dat i on kine t i c s ( Drye r , 1 97 2 ; Drye r

and G las sman , 1 973 ; Drye r , Naege li and G las sman , 1 97 1 ) ha s b een

repor t e d in the li te rature . A ll the expe riment al work had be en

pe rformed on the Pr inc e t on ad i ab at i c , high tempe rature , turbulent

flow re ac t or ( Drye r , S ome re c ent expe riment al wo rk on thi s

re ac t or , alb e i t p re lim inary , and s ome fur the r unde r s tanding of what

i s ne ce s s ary t o mode l c omp lex chem i c al kine t i c sy s tem s are thought

t o be of gre at s igni fi c anc e in fur the r e luc i dat ing the hyd roc arb on

Thi s re s e arch e ffor t was s uppo r te d by the Nat i onal S c i enceDiv i s i on , Re s e arch App l i e d t o Nat i onal Ne e d s , Divi s i on of E ne rgyand Re s ourc e s Re s e arch and Te chno logy , un de r G rant No . AE R 75 - 0 953 8 .

2 8 2

https://www.forgottenbooks.com/join

The un ique advant age s of thi s fl ow re ac t or appr oach should b e

emphas i z e d . By re s t r i c t ing expe riment s t o hi ghly di lut e d m ixture s

of re ac t ant s , and ext ending the re ac t i on s ove r large d i s t ance s,

grad i ent s are such that di ffus i on may be neg le c t e d re la tive t o c on

ve c tive e ffe c t s ( G las sman and E be r s t e in , thus , the me as ure d

s pe c i e pr ofi le s are a di re c t re sul t of chemi c al kine t i c s only .

Thi s type of s p re ad ing i s in c ont ras t t o low pre s sure one d imeh

siona l burne r s tud ie s whe re d i f fus i on e ffe c t s mus t b e de t e rmine d

analyt i c ally be fore us e ful chem i c al kine t i c dat a are ob t aine d .

Whi le the s e flame p roc e dure s have pr ogre s s e d s igni fi c ant ly in the i r

re finement , e s t imat i on of di ffus ive c or re c t i ons remains ve ry

d i ffi c ult .

Furthe rm ore , in the fl ow re ac t or , uni fo rm turbulenc e re s ult s

not only in rapi d m ixing of the ini t i al re ac t ant s , but rad i ally

l - d imens i onal flow charac t e r i s t i c s . Thus re al " t ime " i s re late d

t o d i s t anc e thr ough the s imp le p lug flow re lat i ons . Howeve r , the

re lat i on of a s pe c i fi c axi al c oo rd inat e t o re al t ime i s not we ll

de fine d s inc e the ini t i al t ime c oo r dinat e oc cur s at s ome unknown

loc at i on wi thin the m ixing regi on . One would s us p e c t that ini t i al

m ixing hi s t o ry c ould the re f ore alt e r re ac t i on phenomenon oc curr ing

downs t re am . Howeve r , the exi s t enc e of ve ry fas t e lement ary kine

t i c s , whi ch ini t i at e chem i c al re ac t i on b e fore mixing i s c omple te ,

pe rm i t rapi d ad j us tment of the chem i s t ry t o l oc al c ondi t i ons a s the

flow appr oache s rad i al un i form i ty . Furthe rm ore , the large d i lut i on

of the re ac t ant s and rap i d i ty of the kine t i c s re duc e the c oupl ing o

turbulenc e and chem i s t ry t o the p oint that loc al kine t i c s are func

t iona l ly re lat e d t o the loc al mean flow prope rt i e s ( G l as sman and

E be r s t e in , Thi s c onc lus i on i s al s o support e d expe riment ally

by exc e l lent agre ement of the de r ive d chemi c al kine t i c dat a wi th

that ob t aine d from sho ck tub e s and s t at i c re ac t i on sy s tem s at othe r

tempe rature s . Agre ement als o s ub s t ant i ate s that the re ac t or sur

fac e s d o no t s igni fi c ant ly e ffe c t the gas phas e kine t i c s . Com

par i s on of flow reac t or data from re ac t or tube s of s i gni fi c ant ly

d i ffe rent sur fac e t o vo lume rat i o al s o c o rr ob orate s thi s c onc lu

s i on . Finally and mos t impor tan t , the turbulent flow re ac t or

app r oach pe rmi t s kine t i c s me as urement s in a tempe rature range ( 8 0 0

1 4 OOK ) gene rally inac c e s s ib le t o l ow t empe rature me thod s ( fas t fl ow

E le c t r on Spin Re s onanc e , Kine t i c Spe c t ro s c opy te chnique s , s tat i c

re ac t or s , e t c . ) and high tempe rature te chni que s ( shock tub e s , low

pre s sure po s t flame expe riment s ) .

C ombus t i on of paraff in s ab ove me thane has alway s b een thought

t o be c ompli c at e d by the g re ate r ins t ab i li ty of the highe r alkyl

rad i c al s and by the gre at varie ty o f s e c ondary pr oduc t s whi ch c an

form . The oxidat i on me chani sm charac t e ri s t i c ally fol lows the

Semenov type . Mink off and Tippe r ( 1 96 2 ) have re por te d s ome oxida

t i on me chani sm s of spe c i fi c hyd roc arb ons .

At highe r tempe rature s mo s t have ac c e p te d the primary re

ac t i on in the sy s tem t o b e b e twe en the hyd roxy l rad i c al and the

fue l .

RH i- OH - R +- H20

Re c en t work at Princ e ton ( Drye r , 1 97 2 ) ha s sugge s ted that othe r

r e ac t i ons in add it i on t o thi s one we re imp ort ant ; mamely , in fue l

le an and r i ch c ombus t i on

RH + O —+ R+ OH

and in fue l ri ch c ombus t i on

HH + H — » R + H2

It is inte re s t ing t o revi ew a gene ral pat te rn for the oxi da

t i on o f hydr oc arb ons in f lame s a s g iven by Fris t r om and We s tenbe rg

They s ugge s t two e s s en ti ally the rmal z one s : the primary

z one in whi ch the ini t i al hyd r oc arb ons are at t acke d and re duc e d t o

co, H2, H

20 , and the var i ous rad i c al s ( H, 0 , OH ) and the s e c ondary

z one in whi ch the C0 and H2

are ox id i z e d . The p rimary z one , of

c our s e , is that in whi ch the inte rme d i ate s oc c ur . In oxygen- r i ch

s aturate d hyd r oc arb on flame s , they s ugge s t fu rthe r that ini t i ally

hydr oc arb ons lowe r than the ini t i al fue l form ac c ord ing t o

0 H +~ C H -+ CnH

n 2n+ 2C H C H

2n+ 1'7 '

n— l 2n- 2+

3

Be c aus e hydr oc arb on radi c al s highe r than e thyl are thought t o be

uns t ab le , the ini t i al rad i c al CnH2n+ l

usually s p l i t s of f CH3

and

forms the next ole fini c c omp ound as shown . Wi th hyd roc a rb ons

highe r than C3H8

’ i t i s thought the re may be fi s s i on int o an ole

fini c c ompound and a lowe r radi c al . The radi c al alt e rnate ly s p li t s

off CHE

. The formaldehyde whi ch f orm s in the oxi dat i on of the fue l

and fue l r ad i c al s i s rapi d ly at t acke d in flame s by O , H, and OH , s o

that f ormaldehyde i s us ually found a s a t rac e s ub s t anc e .

In fue l - ri ch s aturate d hydr oc arb on flame s, Fris t r om and

We s tenbe rg s t ate the s i tuat i on is more c omp lex,

although the

https://www.forgottenbooks.com/join

e thene me thane

e thene p r opene

e thene p r opene

e thene p ropene

e thane

2 -me thy l pe ntane pr opene e thene bute ne

e thane

I t would appe ar that the re s ult s in Tab le I would c ont rad i c t e le

ment s of Fris t r om and We s tenbe rg's s ugge s t i on that the ini t i al

hydr oc arb on radi c al CnH usua lly sp l i t s of f the me thy l radi c al .

2n+ 1

I f thi s type of s p li t t ing we re t o oc cur , one c ould expe c t t o find

large r c onc ent rat i ons o f me thane . The large c onc ent rat i ons of

e thene and p ropene found in all c as e s would sugge s t that p rimari ly

the init i al CnH2n+ 1

rad i c al c le ave s one b ond fr om the c arb on at om

from whi ch the hyd rogen was ab s t rac te d . The b ond next t o thi s

c arb on at om i s le s s like ly t o b re ak s inc e thi s type o f c le avage

would re qui re b oth an e le c t r on and hyd r ogen t rans fe r t o f o rm the

ole fin . The ab s t rac t i on o f hydr ogen from a s e c ond c arb on at om

re qui re s ab out kc al le s s fr om the o the r c arb on at om s ( a te r t i

ary c arb on at om re qui re s ab out le s s ) . In a s t rai ght chain

hyd r o c arb on the re are , of c our s e , more hydr ogens on the fi r s t c ar

b on at om s . E s t imat ing re lat ive p r ob ab i li ty o f removal b as e d on

numbe r and e a s e of removal and c ons i de r ing the c le avage rule men

tioned ind i c at e s the prope r t re nds de s ignate d by Tab le I and

2 8 8

re lat ive ly large c onc e nt rat i ons of e thene and p ropene . The s e

re sult s s ugge s t that oxidat i on s tudi e s of e the ne and propene shoul d

b e part i c ularly impor tant .

Figure s 2 - 6 show c le arly that expe r iment ally the re appears

t o b e an ini t i al i s o- ene rge t i c s t ep in the ove rall proc e s s . Of

c ours e , thi s s t ep is not exac t ly is o- ene rge t i c . The c onve r s i on

proc e s s from paraffin t o ol e fin is end othe rmi c ; howeve r , s ome of

the hydr ogen fo rme d during what is e s s ent i ally a pyrolys i s s t ep

d oe s re ac t and re le as e ene rgy . The two re ac t i ons are c ompens at ing

ene rge t i c ally . Thus , i t is b e li eve d that thi s ev i dence sugge s t s

that the re are thre e d i s t inc t , but c oup le d z one s , in hydr oc arb on

c ombus t i on .

1 ) Fol lowing igni t i on , p rimary fue l d i s appe ars wi th li t t le or

no ene rgy re le as e and pr oduc e s uns aturate d hyd roc arb ons and

hyd r ogen . A l i t t le of the hydr ogen i s c onc urrent ly be ing

oxi d i z e d t o wate r .

2 ) Sub s e quent ly,

the uns aturate d c ompound s are furthe r ox id i z e d

t o c arb on monoxi de and hyd r ogen . S imul tane ous ly the hyd r o

gen p re s ent and forme d i s oxi d i z e d t o wate r .

3 ) Las tly , the large amount of c arb on monoxi de forme d i s

oxi d i z ed t o c arb on d i oxide and mos t of the he at re le ase f rom

the primary fie ld i s ob t aine d .

E ach z one mus t have a d i ffe rent tempe rature - rate dependency

and thus at d i ffe rent t empe rature s the impo rt anc e of a given s tep

ab ove may change . Again on the bas i s o f s ome ve ry prelimina ry

expe riment al evidenc e as given by Figu re 7 i t i s pos s ible to put

forth some inte re s t ing s pe c ulat ion s . The ini t i al c ond i t ions of t he

2 8 9

expe riment who s e re s ult s are de p i c t e d in F igure 7 we re s uch that

not only wa s the s t o i chi ome t ry more fue l r i ch than the examp le s of

Figure s 2 - 6 but al s o the ini t i al t empe rature was highe r . E xamina

t i on of F igure 7 reve al s that the maximum c onc ent rat i on of e thene

is found e ar li e r in the sy s t em . E s s ent i ally thi s t re nd indi c at e s

that the exothe rm i c e thene oxi dat i on s t ep ha s b e c ome fas t e r . Thi s

c onc lus i on i s support e d by the fac t that the t empe rature pr ofi le in

Figure 7 ri s e s c ont inually and d oe s not appear f lat ( i s o- ene rge t i c )

throughout mo s t of the p roc e s s as f ound for the c ondi t i ons given

in Figure s 2 - 6 .

The flow re ac t or pe rmi t s hi ghly re pr oduc ib le ac curat e runs

and analy s e s t o be ob t aine d . Al l the dat a point s p re s ente d in

Figure s 2 - 7 are ac tually po int s and not smoothe d dat a . Al though

pre s ent s amp ling t e chni que s p e rm i t only s t ab le s pe c i e s t o b e

me as ure d , e s t imat e s of rad i c al re ac t i on rate s and rat e c ons t ant s

c an b e made . For examp le , s amp le dat a during me thane ( Drye r and

G las sman , 1 973 ) oxi dat i on as de p i c t e d in Figure 8 s hows the

p re s enc e of e thane and the sub s e quent t rans f ormat i on o f thi s e thane

t o e thene . The e thane indi c at e s the p re s enc e of and give s the c lue

t o the me thy l radi c al re ac t i on rate s and c onc ent rat i ons . Furthe r ,

i t i s inte re s t ing t o not e in fue l ri ch , pre —m ixe d e thane oxi dat i on

sy s t em ( Figure 7 ) that ac e ty lene ( e thyne ) c an be i dent i fie d

re ad i ly .

The s e re sult s pe rmi t the c onc lus i on that the turbulent flow

re ac t or is a part i cularly valuab le t ool t o s tudy hydr oc arb on oxi da

t i on pr oc e s s e s .

https://www.forgottenbooks.com/join

1 0 — 1 1

Figure l

2 9 2

03 0

1 0 - 1 2

fl ~

TOTAL CARB ON

Icm 2 : msec(p

TE MPE RATURE

—1 — o0 — 0

— 4o

ar—l"

4 0 5 0 60 7 0 8 0 9 0

I O/ O

4 0 5 0 6 0 7 0 8 0 9 0

DlSTANCE FROM INJE CTIO N ( cm )

I0 6 0

10 4 0

I0 0 0IOO

10 0

N0

103

N

INOEL-JBONVIS

0

1 0 - 1 3

MOLE PE RCENT SPECIES

.0

.0

O O N b m

0

o sssras

N

ea s g 0

F X'

X iKm cx tu o

IG D D O

o

I \ \

at

io

\sD t: o

\b

'

\o

n ou

Ix 1 <1 I b o

O

0 1O U

”!

0 O O O

TEMPERATURE (K)

Figure 3

Chem i c al C omp os i t i on of Spread Propane - Air Re ac ti on

https://www.forgottenbooks.com/join

I cm= msec

TE MPE RATURE

3 0 4 0 5 0 6 0 7 0 80 9 0 IOO

3 0 4 0 5 0 6 0 7 0 8 0 9 0 IOO

D I STANC E FROM I NJE CTI O N ( cm )

Figure 5

Chemi c al C ompo s i t i on o f Spre ad Hexane/Air Re ac t i on

1 0 1 6

2 M E THYLPE NTANE

O ss TE MPE RATUREO\

O

/0

Icm sJIA Imsec D/ D

"

0 IS=0 . | 5 I

CsHe

D/

a /

0 / I-C 4Q /

CH3CH0

o

A(c H CH

4 0 50 6 0 70 80 9 0 IOO llO

DISTANCE FROM INJE CTION (cm)

Figure

Chemi c al Compo s i t i on o f Spre ad 2 - Me thy lpent a ne Air Rea c ti n

1 0 - 1 7

MOLE PE RC E NT SPE C I E S

.0

.0

.0

.O

( D n) .b0 O O

N

O

n)

8 a

TE MPE RATURE ( K )

Figure 7

Chemi c al C ompo s i t i on o f Sp re ad E thane - 02

Re ac t i on

4.

https://www.forgottenbooks.com/join

1 1 - 1

SOME PRE CE PTIONS ON C ONDE NSE D PHASE FLAME

S PRE ADING AND MASS BURNING

I . G las sman

Dep artment o f Ae ro s pac e and Me chani c al S c ienc e sPrinc e t on

N

Unive r sitZPr inc e t on , 0 8 5 O

I . Int r oduc t i on

Inte re s t in pr ob lem s as s oc i at e d wi th fi re s afe ty , pa r tic u

l a r ly a s re late d t o the c ombus t i on charac te r i s t i c s of p las t i c

mate r i al s,has ari s en ove r the pas t year s . Suppo s e dly non- f lammab le

p las t i c mat e r i al s have b e en found not only t o burn but als o t o

emi t re lat ive ly large quant i t ie s of t oxi c c ombus t i on p roduc t s . Ye t

thi s re s ult should not have be en to o s urp ri s ing when one c ons i de r s

that flame s pr e ad wa s the p rimary t e s t c r i t e r i on fo r non- fl amma

b il ity . Mate ri al s wi th sp re ad ing rate s s o l ow that they are

c las s i fie d non— flammab le wi ll burn in fi re s s upp ort e d by othe r

c ombus t ib le s . The phenomena whi ch c ont r ol rate of f lame s p re ad and

rat e o f mas s evolut i on are di s t inc t ly d i ffe rent . The purpos e of

thi s pape r i s t o review c e rt ain fundament al c onc e p t s re lat ed t o

flame s pre ad ing and mas s burning .

I t i s alm os t supe r f luous t o revi ew the fi e ld of f lame s pre ad

afte r the re c ent pub li c at i on of t he exc e llent revi ew by Wi ll i ams

In thi s m o s t c omprehens ive c onc e ptual revi ew Wi ll i am s

t re ate d alm o s t eve ry as pe c t of flame sp re ading d i s c re te and

1 1 - 2

c ont inuous mate ri al s , or i ent at i on , phas e change be fore c ombus t i on,

e t c . In thi s pap e r , only flame s p re ad ing ac ro s s c ont inuous me di a

wi ll be t re ate d . By c ons i de ring the s pre ading p roc e s s only in the

ho ri z ont al ori ent at i on , i t is not ne c e s s ary t o d i s t ingui sh b etween

me lt ing and non- me lt ing mat e ri al . Inde e d be c aus e of the expe ri ence

of the author , the sub j e c t of hori z ontal s pr eading ac r os s li qui d

fue l s wi l l b e revi ewed fi rs t and the ins i ght s gaine d from thi s work

wi ll b e us e d t o c ont ribute t o the unde r s t anding of flame spre ad

ac r o s s s ol i d mate ri al s .

The app roach i s s omewhat di ffe rent than that us e d by Wi ll i ams ,

but thi s sub t le di ffe renc e may he lp s ome in int e rpre t ing the di s

agre ement s whi ch s t i ll exi s t in the fie ld . The autho rs and the i r

c ol le ague s ( Gla s sman , et a l . 1 976 ) re c ent ly revi ewe d the s t at e of

knowl edge of f lame s pre ad ing ac r os s li quid fue ls . What follows are

the b as i c phy s i c al c onc e pt s t aken fr om thi s review wi th c ompari s on

t o the c as e of f lame pr opagat i on ac ro s s s ol i d mat e ri als .

The re lat i onshi p b e twe en the flash po int and bulk t empe rature

of a li qui d fue l de t e rm ine s the type and o rde r of magni tude of the

flame s pre ad . The flash point tempe rat ure i s inde e d a re lat ive

c onc e pt , neve rthe le s s i t pe rmi t s an impo rt ant d i ffe rent i at i on be

twe en two flame s pre ad pr oc e s s e s . When the bulk tempe rature of a

li qui d i s ab ove i t s flash point t empe rature , the re exi s t s above the

li quid fue l a mixture of fue l vap or and ai r that lie s wi th the

flammab i l i ty lim i t s . I t i s gene rally as sume d that equi lib rium

c ond i t i ons prevai l . In the ac tual open c up flash point te s t the

he ight of the small flame igni t i on s ource ab ove the liquid s ur fa ce

s pe c i fie s that at that point when a fla s h i s ob se rved the fue l- a ir

mixture is j us t wi thin the flammab i l i ty lim i t s . The t empe rature

of the liqui d when the f lash oc cur s i s the f lash p oint t empe rature .

It is obvi ous and i t ha s re c ent ly be en shown ( Drye r and Ne uman ,

1 976 ) that the fla sh po int t empe rature vari e s wi th the igni t i on

f lame he ight ove r the li quid s urface . The c lo s e r the s ourc e i s t o

the fue l surface , the lowe r the f lash point t empe rature . Howeve r

a minimum mus t exi s t e i the r b e c aus e at s ome p oint the fue l s ur fac e

exe rt s a quenching e ffe c t . Thus , in the flame s p re ad ing p r oc e s s a

uni que flash point t empe rature for the fue l c annot be s pe c i fi e d ,

but the c once p tual us e i s Obvi ous .

In the c as e a s ment i one d ab ove when the bulk l i qui d t empe ra

ture i s ab ove the fi re p oin t , a flammab le mixture exi s t s e ve rywhe re

ab ove the s urfac e . In the p re s enc e of an igni t i on s our c e,a flame

form s and s pr e ad s ac ro s s the li qui d s urfac e . Unde r the s e t empe ra

ture de fini t i ons the li qui d doe s not c ont r ibut e t o the flame

pr oc e s s . The flame that p ropagat e s i s for all int ent s and purpo s e s

the s ame a s a pre - m ixe d laminar flame . It s ve loc i ty i s ve ry large

due t o the s t rat i f i c at i on of a ir/fue l m ixture ab ov e the s ur fac e

( Feng , e t a l ., The flame c ont inually he at s the c old un

burne d gas e s ahe ad of i t unt i l i t b e gins t o re ac t ( in Wi l l i ams '

c ontext t o an igni t i on tempe rature ) and re lea s e s he at t o c ont inue

the pr oc e s s .

When the bulk t empe rature i s b e l ow the flash p oint , a flame

wi ll s t i ll pr opagate ac r o s s the li qui d fue l , but o the r me chani sm s

mus t c ont r ol s inc e a flammab l e m ixture doe s not exi s t eve rywhe re

ab ove the s ur face . Given that thi s fue l has b een i gni t e d and i s

b urning , the n the re mus t exi s t s ome pr oc e s s whi c h he at s the li qui d

https://www.forgottenbooks.com/join

Fo r a vi s c ous flui d , s uch a s the fue l , unde r a s ur fac e t en

s i on grad ient , i t i s we ll known that

T u ( Bu/By )S

0x

whe re T i s the she e r s t re s s , u the vi s c o s i ty , u the ve l oc i ty

paral le l to the s urfac e , y the di re c t i on no rmal t o the s ur fac e ,

0 the s urfac e tens i on , T the t empe rature and x the di re c t i on along

the surfac e . ( do/dT ) is a phy s i c al charac t e r i s t i c of the li qui d .

One c an re adi ly de duc e the fol l owing p rop o rt i onal i ty for s hal low

pans

( 2 ) Us

l» O'

Xh/LL

whe re uS

i s the s urfac e ve loc i ty and h i s the de pth of li qui d in

the pan . For de e p pool s and othe r analyt i c al c ons i de rat i ons wi th

re s pe c t t o the s urfac e tens i on pr ob l em , one should re fe r t o the

review ( Glas sman , e t a l . 1 976 ) ment i one d e arli e r and the re fe re nc e

the re in .

F rom E quat i on one would e xpe c t that the f lame propaga

t i on ve loc i ty would al s o b e proport i onal t o pan depth and inve r s e l

pr opor t i onal t o the vi s c o s i ty . Inde e d the s e imp ort ant t rends we re

ve ri fie d expe rimentally (Ma cKinven , e t a l . , The expe ri

ment al re s ult s show an almo s t line ar vari a t i on of f lame propa ga t io

wi th i/h wi th s li ght thi ckening of the fue l by a c hem i c al add i t ive

Howeve r i t i s impo rt ant t o ment i on that as the li qui d i s made ve ry

vi s c ous the line ari ty b re ak s down and the flame p ropagat i on

ve loc i ty asympt ot i c ally app r oache s a value of cm/se c . Fo r

c onvent i onal ke ro s ene , the p ropagat i on ve loc i ty i s ab out 3 cm/sec .

The asympt o t i c t rend c ould indi c ate the ons e t of ano the r type of

c ont rolling me chani sm . Inde e d , the pr opagat i on ve loc i ty of

cm/se c i s s im i lar t o that ob t aine d for many s oli d mate ri als

( Fri e dman ,

Not only d o the vi s c os i ty expe riment s vali date the c onc e pt

that c onve c ti on cur rent s in the li qui d are the d ominant he at t rans

fe r m ode , but they als o ve ri fy that he at t rans fe r i s the c ont rol

l ing me chani sm . Inde e d , the p roc e s s e s of fue l vapori z ing , of the

fue l vapor d i ffus ing from the surfac e and mixing wi th the ai r , and

of the flame pr opagat ing thr ough thi s mixture mus t have charac te r

is tic rate s fas te r than the he at t rans fe r rate . In the c as e of

f lame p ropagat i on ac r o s s s ol i d mat e ri al s , whe the r the dominant mode

i s he at c onduc t i on thr ough the ga s o r s oli d , the rate mus t be s lowe r

than the c onve c t ive rate in li quid . Thus i f the c onve c tive rate s

are s lowe r than the othe r s te ps in the le an lim i t p ropagat i on

proc e s s , then inde e d the c onduc t ive s t ep s are . E ven though the

rate of evapo rat i on of s oli d mate ri als are kine t i c ally c ont rolled

whe re a s li qui d s maintain evaporat i on e qui lib rium at the i r surfa ce s ,

the evaporati on rate of s oli d s are re lat ive ly fas t , high tempe ra

ture , high ac t ivati on ene rgy proc e s s e s .

Wi ll i am s ( 1 976 ) de duce s that for the rmally thin solid

mate ri al , c onduc t i on through the gas phas e i s the dominant hea t

t rans fe r me c hani sm and that for the rmally thi ck mate ri a ls , conduc

t ion through the sol id i s the d ominant hea t t rans fe r mechani sm. In

the s pread ing p roc e s s ac ro s s thi ck mate ri a l s , the flame induce s a ir

3 05

c urrent s in the d i re c t i on oppo s i t e t o the p ropagat i on . The e ffe c t

of the s e current s is s t i ll open t o deb ate . Fo r large fi re s rad i a

t i on c an p lay a dom inant ro le in f lame p ropagat i on ac r o s s s olid s,

howeve r rad i at i on i s not ne ar ly a s import ant in l i qui d s (Ma ckinven

e t a l . , C ons i de r ing flame pr opagat i on as a le an fl amma

bility p ro c e s s , pe rm i t s one t o exp lain the e ffe c t s o f flame re t ar

dant s ad de d t o plas t i c s . Inde e d , one should again re c all , that

flame re t ardant s lim i t flame s p re ad , but re t ar de d mate ri al s wi ll

burn . The thre e mo s t c ommon means o f re t ard ing flame pr opagat i on

are t o ad d ch lo rine ( o r o the r halogens ) , ant imony o r pho s phor ous

c ompound s t o the p olyme r s t ruc ture . It i s we l l known that halogen

affe c t ( narr ow ) flammib il ity lim i t s . The pre s enc e of a hal ogen in

the polyme r would re qui re the polyme r t o b e heate d t o a high

tempe rature be fore a flammab le mixture c ould b e c re at e d due t o the

p re s ence of the chlor ine at om . Ant imony i s found t o b e e ffe c t ive

only when in halogenat e d c ompound s . Ant imony chlo r i de i s a gas e ou

c ompound and i t appear s that the role o f the ant im ony i s t o fac i l

it a te the p re s ence of ch lor ine at om s in the gas phas e . In c ont ras

pho sphor ous alt e r s the s urfac e c harac t e ri s t i c s of the p olyme r ,

c aus e s a me lt and e ffe c t ive ly inc re as e s the heat of gas i fi c at i on .

C ondens e d phas e s mus t burn as d i f fus i on flame s and the flame

mus t b e e s s ent i ally at the s t oi chi ome t ri c mixture rat i o . Inhib i

t or s such as halogens are on ly e ffe c t ive at the flammab i li ty lim i t

whe re the rad i c al s affe c t ing the chain pr opagat i on are s c arc e . In

s to i chi ome t ri c flame s,

rad i c al s are abundant and any removal by

inhib i t i on is ine ffe c t ive in alt e r ing the mas s burning pr oc e s s .

Thus mate ri al s wi th flame re t ar dant s wi ll alt e r the rate of f lame

3 0 6

https://www.forgottenbooks.com/join

( 3 ) ofmf/4w r2 ( pp/r s )

ln ( l+ B ) ( A/Cpr ) 1 n ( l+ B )

(u/r ) ln ( l + B )

whe re Gfis the mas s f lux g/se c cm

2, m

fthe mas s burning rate g/se c ,

r the part i c le rad iu s , B the t rans fe r numb e r , and D , A, C u ,s p ’

and p the normal phy s i c al p r ope r t i e s . E quat i on (3 ) i s ob t aine d

unde r the a s sump t i ons that a quas i s t e ady s t at e exi s t s and the

part i c le i s l ike a por ous s phe re fe d w i th fue l at a rat e e qual t o

the c ons umpt i on rate , Le l and c ons tant phy s i c al pr ope r t i e s . The

t rans fe r numb e r c an t ake any of the fol lowing form s , al l of whi c h

are e qual

( 4 ) B - mf s)

( 5 ) B U&JT;

— T l ) )

( 6 ) B ( Cp( T

w

- TS)

whe re mo

and mf

are the mas s frac t i on of the oxi d i z e r and fue l

re s pe c t ive ly ; T , the t empe rature , H , the he at ing value o f the fue l

in c a l/gm; LV

’ the latent he at of evap orat i on ; i , the mas s

s t oi chi ome t ri c index ; and the s ub s c r i pt s and m re fe r t o th e c ond i

t i ons at the sur fac e and in the amb i ent atmo sphe re re s pe c t ive ly

E quat i on (3 ) may be in t e rpre t e d in te rm s of an ac t ual drop le t

burning , i . e .

dm/d t( 7 ) —m

( e )

C omb ining E quat i ons (3 ) and one ob t ains

3 0 8

1 1 - 1 0

( 9 ) drg/d t ( 2Dp/pl ) ln ( l+ B )

The right hand s i de of the E quat i on ( 9 ) i s c ons t ant and thus the

rate of change of r2

wi th t ime is c ons tant . Thi s re sult,

of

c our s e , c or re spond s t o the s o- c alle d d2( or r

2) law that i s found

expe riment ally .

I t is int e re s t ing t o note that the s phe ri c al par t i c le burning

in a qui e s c ent atmo sphe re i s the only mathemat i c ally t rac t ab le

pr ob lem . The one d imens i onal burning of a s t rand of fue l or pool

of li quid is not mathemat i c ally t rac t ab le unle s s one as s ume s that

at a fixe d d i s tan ce , say 6 , ab ove the sur fac e amb ient c ondi t i ons

exi s t . In thi s c as e , re fe rre d t o a s the s t agnant fi lm c as e , i t i s

re ad i ly shown that

( 1 0 ) of ( B p/o ) ln ( 1 + B )

A burning po o l of li qui d o r a v olat i le s ol id wi ll e s t ab li sh

a s t agnant fi lm he ight due t o the natu ral c onve c t i on whi ch ensue s .

From analog ie s t o heat t rans fe r wi thout mas s t rans fe r , a fi r s t

appr oximat i on t o thi s li qui d pool burning p rob lem may be wri t ten

( 1 1 ) d /u ln ( l+ B ) a Gra

whe re G r i s the G rasho f numbe r ; d is the d i ame te r of pool o r

s t rand , a e qual s for lam inar c ond i t i ons and for turbulent

c ond i t i ons . I f ai r i s f orc e d c oncent ri c ally around the pool or

s t rand , ve ry much like the B urke - Schuma nn gas eous fuel je t p roblem ,

then again , one c an as s ume a s t agnant fi lm p rob lem .

Whe n the c onve c t ive flow of ai r i s norma l or opposed to the

mas s evolv ing from the surfac e , the solut ion i s more complex and

1 1 — 1 1

the s t agnant fi lm analy s i s d oe s not hold . E mmons ( 1 956 ) s olve d

the problem of a burning longi tud inal surfac e wi th fo rc e d c onven

t i on . The fue l is e s s ent i ally a flat p late wi th a le ad ing e dge .

The p rob lem is a l s o de s c r ibe d by Wi ll i am s ( 1 9 65 ) and i s s im i lar t o

the Blas ius pr ob lem for the growth o f a b oundary laye r ove r a f lat

pl ate . The E mmon s re sult f o r Prand t l numb e r e qual t o one t ake s the

form

( 1 2 ) G,

f ( o ) ]

efx/u Re}

/2T

whe re ReX

i s the Reyno ld 's numb e r b as e d on the d i s t anc e x f rom the

le ad ing e dge of the longi tud inal fue l s ur fac e and f ( o ) ] i s a

Blas ius type variab le whi ch i s a func t i on of the t rans fe r B .

Wi ll i am s ( 1 965 ) give s the g raphi c al re lat i on be tween f ( o ) ] and

G las sman ( 1 977 ) has shown emp i r i c ally that

( 1 3 ) f ( o ) l

ove r a large range of B value s .

I t would appe ar t o f o ll ow that dat a fo r s phe r i c al par t i c le s

burning in a c onve c t ive atm o sphe re c ould c or re late a s

( 1 4 ) f ( Rei/2>

whe re Rer

i s the Reynold ‘ s numb e r b a s e d on the drop le t . E ve n

though a wake may exi s t in whi c h ve ry li t t le burning oc cur s ,

Spald ing ( 1 9 55 ) ha s s hown that E quat i on ( 1 4 ) wi thout B c or re

late s data re lat ive ly we ll .

https://www.forgottenbooks.com/join

1 1 - 1 3

b e c aus e the e quat i on has deve l ope d in the framework of a d i ffus i on

analy s i s , and E 1 me ans that the s oli d i s gas i fi e d by the rad i

ant f lux al one .

As ment i oned e ar lie r , current ly the re are many inve s t igat or s

s eeking t o e s t ab li sh te s t s whi c h de t e rm ine the mas s burning rate

of p las t i c s . One of the be s t of the s e pr oc e dure s is tha t given by

Tewa r s on and Pi on In the i r expe r iment , an 02/N2 m ixture

pas s e s around the burning p las t i c and the rat e of f low i s he ld

c ons t ant . The ga s f low is c onc ent ri c wi th the c i r cular cy linde r

s amp le and holde r and i s in the s ame d i re c t i on as the mas s evolu

t i on fr om the gas i fying s amp le . The gas fl ow i s c ont aine d wi thin

a quart z c i r c ular cy linde r . Radiant he at e r s out s i de the quart z

cy linde r pe rm i t an ext e rnal flux t o b e impo s e d on the s amp le .

Tewa r s on and Pi on re p ort s ome exc e l lent dat a and one of the int e r

e s t ing find ing s is that a line ar re lat i on exi s t s b e twe en the mas s

burning rate of the p las t i c and the mas s f rac t i on of oxygen in the

fre e s t re am , mw

. The lineari ty b re aks down at highe r value s0

of mom

. Glas sman ( l977 a ) ha s at t empte d t o explain the s e re s ult s

by arguing that the o rde r of B and [ B / ( l - E ) ] mus t b e much le s s th

one and that inde e d the B value s f or mo s t plas t i c mate ri al are

smalle r than previ ous ly e s t imat e d . For s uch small value s

( 1 7 ) - E ) ) s B / ( l — E )

From E quat i on ( 1 5 ) then

( 1 8 ) mfat — E h

Sub s t i tut ing E qua t i on ( 1 6 )

1 1 - 1 4

( 1 9 ) mf

( A/Cp6 )B + Q

R/L

v

Taking the fo rm of B given by E quat i on ( 6 )

( 2 0 ) - ( Cp(T

w

E quat i on ( 2 0 ) shows the l ine ari ty wi th re s pe c t t o m0

CD

and 6 pr ob ab ly do vary s ub s t ant i ally fo r a fixe d c onve c t ive

c ond i t i on and vari ous mate r i als , i t is po s s ib le t o pe rfo rm experi

S inc e A,G )

ment s with l i quid s of known value s of B t o de te rmine ( k/CPO ) from

a plot of mfv s . B or more c orre c t ly m v s . Then from a

f

me asure d mf

of a p las t i c , it s B value can be de t e rmine d .

Tewa r son ( 1 977 ) ha s shown for plas t i c s that the te rm

Cp( T - T

S) i s a re lat ive ly large ne gat ive numbe r and c annot be

igno re d in c ompar i s on t o imOwH a s i t i s oft en done for li qui d s .

Thi s fac t and large value s of LV

c ont ribut e t o making E value s o f

plas t i c s smal l .

I t i s d i ffi cult t o de t e rm ine whe the r the non- l ine ari ty o f

Tewa r son 's m

mp lo t s b re ak s d own due t o the fac t that at hi ghe r m

o

value s B may not be small . Tewa r s on ( 1 977 ) report s that for large r

value s of m00°

b lack char fo rmat i on on the surfac e of the PMMA wa s

found . Such change s in pyrolys i s me c hani sm c ould , of c ourse ,

c aus e the ob s e rved t rend s .

In de aling wi th c harring c e llular plas t i c s the mas s evolution

and burning proc e s s appear s t o be di ffe rent . The s e mate ri als a re

d i f fi c ul t t o " burn " exc e p t unde r ve ry large exte rna l ra d i ant

fluxe s . Unde r suc h rad i an t fluxe s , the re i s the ini t i al evolut ion

of re lat ive ly large amount s of combus t ib le gase s whi c h wi ll burn

1 1 — 1 5

when an igni t i on s ourc e i s p re s ent . The rate of evoluti on of

gas e s de c re as e s wi th t ime a s the py r oly s i s gas e s mus t c ome fr om

gre ate r de p ths wi thin the c e llular p las t i c . It i s the s e gas e s

whi ch c ont ribute t o r oom fi re s and are pr ob ab ly a fac t o r in the

s o— c alle d flashove r pr ob lem .

The burning of the char c e rt ainly would not c ont ribute in

the e ar ly s t age s of fi re , b ut the char wi ll burn Ve ry much like a

por ous c arb on ( o r c oal ) par t i c le . In a maj or fi re the s ur fac e

tempe rature of c har woul d be high enough s o that the s urfac e oxida

t i on of the char was k ine t i c ally fas t and c ont r olle d by the diffu

s i on o f oxygen t o the sur fac e . Unde r the s e c i r cum s t anc e s i t i s

again inte re s t ing t o ob s e rve that

( 2 1 ) m A . ln ( l + B )f

whe re m would be the mas s burning rate of the char . I f E quat i onf

(4 ) i s us e d as the form of B , then a s imp le exp re s s i on re s ult s

s inc e for thi s type of d i ffus i on c ont r ol le d he t e rogene ous s ur face

burning mfs

O and

( 2 2 ) B II Ho

5‘

For burning in ai r im is small wi th re s pe c t t o one , and again0

s ince ln ( l+ B ) g B for small B , one has that

( 23 )

the s ame re s ult that one ob t ains f or the po r ous c arb on ( c oal )

par ti c le as Glas sman ( 1 977 ) has re c ent ly di s cus s e d .

https://www.forgottenbooks.com/join

1 1 - 1 7

Glas sman , I ., S ant oro , R . J . and Drye r , F . L .

"A Revi ew and

S ome Re c ent Re s ult s of the Prince t on Pr og ram on Flame SpreadOve r C ondens e d Phas e C ombus t ib l e s ,

" p re s ente d at the FallMe e t ing of the We s t e rn S t ate s Se c t i on of the C ombu s t i onIns t i tute , La Jol la , C ali f . Pape r No . wsc1 - 76 - 2 9

G la s sman , I .

”C ombus t i on , Ac adem i c Pre s s , New York .

Glas sman , I . ( l977 a ) .

" C omment on 'Flammab i l i ty o f Plas t i c s .

I . Burning Inten s i ty ' by A . Tewa r son and R . F . Pi on ,

" t o appe arin C ombus t i on and Fl ame

Kanury , A .M .

" Int r oduc t i on t o C ombus t i on Phenomena ,

Gord on and B re ach , New York .

Ma ckinven , R . , Hans e l , J . , and G las sman , I . C omb . Sc i andTe ch . 1 , 2 93 .

Sp alding , D . E .

" Some Fundament al s of C ombus t i on ,

"

But te rwor ths , Lond on .

Tewa r son , A .

" Rep ly t o C omm ent s on Our Pape r , to appe arin C omb us t i on and Flame .

Tewa r s on , A . and Pi on , R . F . C omb . and Flame 2 6 , 8 5 .

Wil li am s , F . A .

" C ombus t i on The ory,

Add i s on - We s ley,

Re ad ing , Mas s .

Wi lli am s , F . A .

" Me chani sm s o f Fi re Sp re ad ,

" invi t e d pape rS ixte enth Int 'l ) Sympo s ium on C ombus t i on , C amb r i dge , Mas s .

3 1 6

Thi s r epor t wa s pr epar ed a s an ac c ount o fGov ernment spon so r ed work . Ne it her t he

Unit ed St at e s,nor t he Admini st r at ion ,

nor any p er son ac t ing on behal f o f t he

Admini st rat ion

A . Make s any war rant y or r epr e s ent at ion,

expr e s s o r impl i ed , wit h r e sp ec t t o t he

ac c urac y , c ompl et ene s s,or u s e fulne s s o f

t he informat ion c ont a ined in t hi s r epor t ,

or t hat t he u se o f a ny informat ion,

apparatu s , met hod , or pr oc e s s d i s c l o s edin t hi s r epor t may not infr inge pr ivat elyowned r i ght s ; o r

B . A s sume s any liabil it i e s wi t h r e sp ec t t ot he u se o f , or for damage s r e su lt ing fromt he u s e o f any informat ion

,appara tu s

,

me t hod,or p r oc e s s d i s c lo s ed in t hi s

r eport .

As u s ed in t he above,

" per son ac t ing on behalfo f t he Admini st rat ion " inc lud e s a ny emp lo ye eor c ont rac t or o f t he Admini st rat ion , oremploye e o f suc h c ont rac t o r

,t o t he ext ent

t hat suc h emp loyee o r c ont rac t or o f t he

Admini st ra t ion ,or employe e o f suc h c ont r ac t o r

pr epar e s,d i s s eminat e s

,or provid e s ac c e s s t o ,

a ny informa t ion pur suant t o hi s employment o rc ont r ac t wit h t he Admini st ra t ion , or hi sempl oyment wi t h suc h c ont rac t or .

This book m ay be kept NOV 2 1978

FO U RTE E N D A Y S

A fi ne W illbe charged for each day the book IS kept ovemm e

https://www.forgottenbooks.com/join