CRANFIELD
ON THE SUDDEN CONTACT BETWEEN
A HOT GAS AND A COLD SOLID
by
1 REPORT NO.124 Januajcy, 19^0 C O L L E G E O F A E R O N A U T I C S C R A N F I E _L_D On t h e sudden c o n t a c t
between a h o t gas and a c o l d s o l i d
"by
-J . F . C l a r k e , B . S c , P h . D . , A . P . R . A e . S .
SUMtARY
The flmf i n d u c e d by t h e sudden c o n t a c t between a s e m i - i n f i n i t e esqjanse of gas and a s o l i d , i n i t i a l l y a t d i f f e r e n t t e m p e r a t u r e s , i s examined on t h e b a s i s of a l i n e a r continuum t h e o r y . F o r t i m e s l a r g e compared ¥d.th t h e mean time bet\TCen molectilar c o l l i s i o n s i n t h e g a s , t h e -velocity and p r e s s u r e d i s t i j r b a n c e s a r e found t o be c o n c e n t r a t e d a r o u n d a vra-ve f r o n t p r o p a g a t i n g out from the i n t e r f a c e a t t h e ambient i s e n t r o p i c sound s p e e d , v d a i l s t , neair t o t h e i n t e r f a c e , t h e s e d i s t i x r b a n c e s axe s m a l l and t h e gas teirrperatures a r e n e a r l y e q u a l t o t h o s e p r e d i c t e d b y t h e c l a s s i c a l c o n s t a n t p r e s s u r e h e a t c o n d u c t i o n t h e o r y .
The p o s s i b l e s i g n i f i c a n c e of t h e s e r e s u l t s i n c o n n e c t i o n v/ith r e f l e c t e d shock wave t e c h n i q u e s t o measure h i g h t e m p e r a t u r e gas p r o p e r t i e s i s commented upon,
Summaiy
L i s t of Symbols I n t r o d u c t i o n The Equations The Problem
Laplace Transform Solutions Solutions for cr =. ^
5.1, Evaluation of I^ 9 5.2. Evaluation of T,, 12 5.3» The Temperature at Large Times 17
5.if. The Pressure and Velocity Perturbations 19 5.5. Interface Teniperature and Conditions
at Zero Time 21 Solutions for o" = 0 23 6 . 1 . Ö ( x , t ) for X < t / v V a n d t large 25
6 . 2 . 6 ( x , t ) for X > t / v V a n d t l a r g e 2? 6 . 3 . Pressure and Velocity Disturbances "29
Conclusions 30 References " 32
LIST OP SYMBOLS a o % "n erf
h
P
4
Q
R
st
T
u
X ye
KX
^ V p <T T SuffixesIsentropic soimd speed Isothermal sound sx)eed
Specific heat at constant pressure
Wober parabolic cylinder function of oilier n Error function
Specific enthalpy Pressure
Energy flux vector
Ratio of thermal properties Gas constant per unit iTiass
Specific entropy or Laplace transform variable Time
Gas temperature Velocity
Distance
Ratio of specific heats
Temperature difference, T - To,, Diffusivity Conductivity Viscosity Kinematic viscosily Density Prandtl number Shear stress tensor
" Initial conditions in the gas m Refers to value in the solid
1 , iS'^^C^HSJ'ASG
The condviction of heat i n a coiupressiblo gas w i l l i n general oe acoonipanied by changes i n the gas i^ressuiv, density and veloc-ii^-. I t i s the piurpose of ttie p r e s e n t work t o study these changes f o r tlie . . p a r t i c u l a r case of the sudden contact betv/uen a semi-ini'inite sol3.d and
s e m i - i n f i n i t e g a s , i n i t i a l l y a t d i f f e r e n t uniform temperatures. This simple t h e o r e t i c a l model would be d i f f i c u l t to achieve i n p r a c t i c e , b u t something approachin.g t h i s s i t u a t i o n i s found \-/iien a shock wave r e f l e c t s from the closed end \;'all of a conventional shock t u b e ,
and i t i s hoped to gain some idea of vAiat may happen i n t h i s nioi-e
complicated case from the p r e s e n t stvidy» The i n t e r : ; s t i n r e f l e c t e d shock wave zones a r i s e s from the ease vidth v/hich a saiiiple of gas a t a higli
temperature can be produced by t h i s p r o c e s s , and tlie resialtii:.g p o s s i b i l i t y t h a t measurements of the gas p r o p e r t i e s under these conditions ca:i then be made.
The question of tlie c o m p r e s s i b i l i t y e f f e c t on heat conduction hajs been examined previously by Cole and Wu (1952) for the case of the Dirac h e a t p u l s e , but these w r i t e r s have made tlie assuinption t h a t gas v i s c o s i t y can be neglected. I t Tivill be shown belovT t h a t v i s c o s i t y ccn be included, hovTOver, provided t h a t P r a n d t l number equals 3/4. Like C o l e ' s aiidV/u's, the p r e s e n t treatment i s based on the assumption of small distiirbaiioos, so t h a t l i n e a r equations can be derived. To avoid over-coiriplication a t t h i s stage the gas i s assumed to c o n s i s t of s t r u c t u r e l e s s p a r t i c l e s , i . e . to be monatomic and unexcited e l e c t r o n i c a l l y , and t o be p e r f e c t both thoriTially and c a l o r i c a l l y . Althougli P r a n d t l number equrJ- to 3/4
i s a most p r a c t i c a l s t a t e of a l T a i r s , and the s o l u t i o n s are quite readily obtained i:i t h a t event, the zero P r a n d t l number case i s also exa:ained here i n aii attempt to assess how di-astic tlie zero v i s c o s i t y assunrption w i l l b e . The gas i s t r e a t e d as a continuum, and, since the c h a r a c t e r i s t i c
time for the processes to be studied turns out to be comparable v-dth the mean time bet\".tjcn molecular c o l l i s i o n s , we are e f f e c t i v e l y liiiiitcd t o a consideration of "large time" s o l u t i a i s only.
Reference w i l l be me.de i n l a t e r s e c t i o n s to the " c l a s s i c a l solution" of the sudden contact problem. This s o l u t i o n t r e a t s the gas as a s o l i d and assiJiiies a t the o u t s e t t h a t only i t s temperatiare w i l l change subsequent t o the i n i t i a l i n s t a n t , the pressure ( o r density) rcraainirig c o n s t a n t . Gas v e l o c i t i e s are a l s o assimod t o be zero throughout. I t i s one of
the r e s u l t s of the present a n a l y s i s t h a t the c l a s s i c a l constant pressure s o l u t i o n i s approached asyinptotically -with i n c r e a s i n g tiirse i n tlio
regions n e a r t o the i n t e r f a c e . To a s s i s t i n the i n t e r p r e t a t i o n of these r e s u l t s , a sketch of the c l a s s i c a l s o l i d - t o - s o l i d contact tentperature p r o f i l e s i s given i n P i g . 6 .
2
-The gas is assumed to be tliermally and calorically perfect so that its pressure p, density P temperature T and specific enthalpy h are related as follows,
p. = PRT ; h = C T . (l) R is the gas constant for unit mass and C the (constant) specific heat
ir
a t constant p r e s s u r e .
The c o n t i n u i t y equation i s
(•vdiere D/Dt = 9 / 9 t + u9/®x for the one-dimensional unsteady problem) , b u t a more convenient form for p r e s e n t purposes can be derived by w r i t i n g f i r s t of a l l
DP fdp\ Dp ^ fBp\ Ds /-N
Dt = V^;^
m ^ KTsJ
Dt
• (^)
s is the specific entropy and
Hi - ^" •
^^'
where a is the usual isentropic sound speed. The derivative (9p/9s)
P
is readily evaluated for a gas with the simple thermodynamics described in eq,(l) and we find that
V /p
( y i s the (constant) r a t i o of s p e c i f i c h e a t s ) . Writing q for the heat
flux and r for the viscotis p a r t of the s t r e s s tensor, the energy equation i s
^^p Dt Dt - ^ ^ ^ • ^^^ Conibined \'7itli the thermodynamic equation Tds = dh -P^^dp, eq. (6) shows t h a t
It follov/s at once that the ccaatinuity equation can be fo-itten as
2| . p a ^ ^ . ( y . 0 ( | | - r | ^ ) = 0 . (8)
The heat tlux and viscou.s stress are assimied to have their usual vaJ-iies
^ '^ ax ' ^ - 3 3x » ^^^
so that eqs, 6 and 8, covcpled with tlie momentum equation
constitute three equations for the vinknowns p , u and T.
\7G shall HOT/ assume that all of these three unloiown quantities differ but little from their tindisturbed va.luGs, the undisturbed state being defined as one of unifonn pressure p ^ and temperature 1^ ,
and zero velociiy, over the vAiole of the region of interest. Then the equations can be linearised by neglecting all terms involving squares or products of disturbance quantities, leading to the follovdng tliree equations,
•= 0 , (11)
(12)
(13)
Since we s h a l l be i n t e r e s t e d in problems for which boundsir^' values are expressed mainly i n terms of tenrncrature, a single equation s a t i s f i e d by T a3.one T i l l be derived from e q s . 11 to 1 3 . The thermal d i f f u s i v i t y K and kinematic v i s c o s i t y v are defined as
OO CO / . . \
" = p ~ ^ J t^ = — , (14)
CO p CO
and the P r a n d t l nuiribcr a* as
S2 8t ^ at 00 p + + aT at p CO ax -2 9 u / ^ 9 ^ ^ y -L 9^u 3 CO g ^ 2 | E - X ^ d t °° 9 2 1) X ê!l^ - 9x' = 0 , = 0 . V = a- K , (15)
4
-The equation s a t i s f i e d by T i s then 1 9fT %. 9t^ J.
at
a^T 9x" <J K3a4
9lS
ax^ 9x" klJljtJ. a'^T _ a^T 3 t ax^ = 0 (16) 3 . The ProblemAt t h i s stage i t i s convenient t o formulate the a c t u a l probleii t o be t a c k l e d . The g a s , whose temperature i s t o s a t i s f y eq. 16, i s assumed
t o occupy tlie half-plane x > 0 and t o be a t r e s t a t uniform rjressure p^ and temperature T^ for a l l t .< 0. At time t = 0 a s e m i - i n f i n i t e s o l i d , Tdiich has been a t a imiform temperature T for a l l t < 0 , i s placed i n
contact with the gas along the plane x = 0. The s o l i d then occupies the h a l f - p l a n e x < 0. Without any l o s s of g e n e r a l i t y T can be s e t equal t o z e r o . Subsequent t o time t = 0 the temperature T of the s o l i d i s assumed t o s a t i s f y the c l a s s i c a l h e a t conduction eqioation
aT m 9t a^T - K 9x = 0 (17)
•vidiere K is the appropriate diffusivity, (assumed constant here), A new temperature Ö is defined for the gas such that
6 = T - T^ . (18) The initial conditions then become
T = 0, t < 0,
m ' ' X < 0 6 = 0, t < 0 , x > 0 , (19)
Compatibili-ty of temperature ajid heat flu2C a t the i n t e r f a c e reovdre
aT. 96
T = 6 + T , t > 0 , x = 0 J m-T^ = ' ^ ~ , t > 0 , x = 0 ,
m 9 X 00 9 x
.(20) where \ i s the (constant) thermal conductivi-ty of the s o l i d , and tvro
f u r t h e r conditions are
A further requirement is that tlic gas velocity u shall be zero at x = 0 for all time, since the solid is ii:pernieable. This condition can be translated into a temperatui-e condition at x = 0 via eqs. 11 to 13
by eliminating p and all derivatives of u which contain operatiais involving 9/9x in terms of T (or Tdiat amounts to the same thing,6) , leaving an
expression for d^v/dt^ in terms of derivatives of 6 . Then, since u = 0 Y*ien X = 0 and t > 0 we have also a^u/9t^ = 0, x = 0, t > 0 and it follovra that
9x9t " , 2
9x^at 3a2 axat^
0, X = 0, t > 0
.... (22) The conditions 19 to 22 inclusive are sufficient to specif^- the problem. Before proceeding Tith a solution, hov/ever, the equations Tn.ll be written in dimensionless form according to the dejfinitions
t = Kt' X = /CX' (23)
Then we have t o solve
a t ' _9fe ^ 9^6 ^^a 9^6_ a t ' 2 ax'^ ^ 9x'^ ax'2 and 9Tr. / K \
of "
\KJ
subject t o the conditions
[( y
a^T m 9x'"L r(y.z.o-/3) ^ - ^ " 1 =0 (24)
'2 L a t ' 2 9x'^ J = 0 T^^ = 0 , f < 0 , x' <0 ; 6 = 0 , t ' < 0, x' > 0, T = 6 + T , f > 0 , x ' = 0 ; m CO ' ' m 9T m 9x' 96 (25) (26) m ' g ^ , t ' > 0 , X ^ = 0 , (27) , f > 0 ; e . O, x ' - + 0° , f > o , (28) a^e ^^yo- B"^ e dx!dbf 9 x f ^ a f 3^6 9x^' + M £ ö _ = o, ^^0^ t'> O . ax' a f (29)M. o •*
In subsequent sections we shall omit the primes from x and t
for brevity, since from no?/ on TWD shall work exclusively in the dimensionless co-ordinates.
!(.. Laplace Transform Solutions
The Laplace transform Tïith respect to the dimensionless time t will be denoted by a bar (-) over the appropriate symbol, e.g.
6 (x J s) = j 9(x, t) exp(-st)dt, o
(Entropy will not be needed in subsequent discussions so that from now on s refers to the transform variable),
Y/ith conditions 26 (expressing initial quiescence) the operational forms of eqs. 24 ancL 25 are
-(iv) , .^ _
6 (1 + ^ s) - 6" (3 ^ (y J£)^^) + 33 6 = 0 , (50)
T " - ( S K/K ) T = 0 . (31)
(Primes denote differentiation Td.th res^pect to x ) ,
Eq. 31 can be solved at once^with the appropriate condition from eq,28, to give
1^ = A(s) e x p ( ( s ^ K j ^ x ) , (32)
Tdiere A ( S ) is a function of s to bo found from the boundary conditions, Conditions 27 and the second of 28 in transform form are
6 (0 J s) = A ( s ) - T ^ s " ' J A Q /s = 6'(0 j B ) , (33)
yiiere T/e have TiTitten
v J Ï
- ^- (^^'
' m
The transform versions of the remaining conditions28 and 29 are
é ( x ; s) •» 0 , X •» OS , ( 3 5 )
and the problem i s now reduced to t h a t of finding a s o l u t i o n of eq. 30 subject t o conditions 33» 55 and 36.
/in appropriate solution of eq. 30 i s 6 oc exp(a x) where a i s any one of the four r o o t s of tlie axjxiliary b i q u a d r a t i c equation
(1 + k-ya- s/3)a^ - s ( l + (y + W 3 ) s ) a ^ + s ' = 0 , (37)
The general s o l u t i o n of t h i s equation could be Tïtitten dovm, but would give formidable
s p e c i a l o a s e s ,
give formidable values for the a , Instead we s h a l l consider tvTO
( i ) cr = i
Ylhen
s o l u t i o n s
4 •
c = •£ , eq. 37 f a o t o r i s e s quite simply and gives the fOTJT
a = ± VT J ± s ( l + y s ) " ^ (38)
Condition 35 excludes the solutions with positive signs and it follovrs that the most general solution of eq. 30 subject to this requirement is
6 (x ; s) = B ( S ) exp(- s^x) + C(s) e3cp(- s(l + ys)'^x). (39)
The value T for c is not far from the accepted value for a number of interesting gases, air for exainple for v/hich cr = 0.72 is quoted, so that the solution 39 should give a plausible description of the phj'sical picture.
(ii) CT = 0,
This not very p r a c t i c a l value of the P r a n d t l number corresponds t o the s o l u t i o n for which K i s assumed t o have a s u i t a b l e f i n i t e , non-zero vaiue w h i l s t the v i s c o s i t y ^ i s s e t equal to zero. Physically'-, of c o u r s e , t h i s i s quite inadmissable b u t i t i s argued t h a t -the e f f e c t s of heat
•conduction and -viscosity arc s i r r i l a r , so t h a t a reasonable p h y s i c a l p i c t u r e should be obtained by ignoring one of -them a l t o g e t h e r . This i s ra-ther l i k e saying t h a t P r a n d t l number i s of order u n i t y so t h a t -vve s h a l l approximate to i t s e f f e c t by p u t t i n g i t equal to zero '. - b u t -there does seen t o bo an i n t u i t i v e f e e l i n g thf.t the pliysical p i c t u r e should be
r e t a i n e d despi-te -this. Accordingly we s h a l l examine -the a = 0 case v/i-th •this i n mind. As remarked i n -the I n t r o d u c t i o n , Cole and Wu have studi.ed the Dirac heat pulse problem f o r G" = 0 and Lagerstrora, Cole aiid T r i l l i n g (1949) ha-ve studied a v a r i e t y of esson-tially viscqus problems voider the
8
-assun55tian \ = O while r e t a i n i n g / j fini-te and non-zero. (As aan be seen from the equations 1 1 , 12 and 1 3 , X = 0 uncouples -the ' p , u* problem from the energy equation, so -that the present theory i s not d i r e c t l y com^parable with Lagcrstrom's. One might say -tliat T/e are interes"ted i n problens
p r i r a a x l l y of hea.t conduction).
T/hen o" = 0 -then, eq, 37 has "the s o l u t i o n s
a = ± Vs* (1 + y s ) / 2 ± (1 + y s ) V 4 - s . (40)
The -two solutions starting + -/"s etc, must be abandoned to confoim with eq, 35 and so, for our purposes, -vve have
ê ( x J s) = B' ( s ) exp f - Vé']^ (1 + y s ) / 2 + | (1 •<• y s) / 4 ~ + C' ( s ) exp F - ^ré'\ (1 + y s ) / 2 - ["(1 + y s ) V 4 - s
J X
( 4 1 ) ' (The constants B' and C' are d i f f e r e n t from B and C i n eq. 3 9 ) . Cole and Wu remark tliat s e t t i n g /J= 0 s i m p l i f i e s blie equations t o be studied, Examination of eq. l6 would c e r t a i n l y tend t o sug^^est -that t h i s i s t r u e , but comparison of the s o l u t i o n s 39 and 4 ' i n d i c a t e t h a t the reverse i s -the c a s e , c e r t a i n l y when n i s r e t a i n e d and c put equal t o 3/i •
Port\ma-tely Cole and Wu were able t o find a t r a n s f o r a a t i o n iThicli renders an at-tack on -the o" = 0 case p o s s i b l e , b u t , as w i l l become evident
below, i t must be applied wi-th some care and grea-ter labour i s invol-ved i n -the c = 0 problem -than Tdien cr= 3 A .
Neither case produces a p a r t i c u l a r l y simple soluticm owing t o -tlie appearance of the coiTiplica-ted exponential f u n c t i o n s , so i t may be advisable t o examine b r i e f l y the pliysics of the sitviation i n order t o decide j u s t what kind of s o l u t i o n s i t would be b e s t to aim f o r . From equations 23
i t can be seen -that tlie c h a r a c t e r i s t i c time and length for the sys-tem are
n/a. and '^/s^ respecti-vely. Simple k i n e t i c -theory indica-tes t h a t
X - (1/3) PÖ&C , where c i s -the mean molecular speed, ^ -the imjan free pa-th and C -the constant volume s p e c i f i c h e a t . Consequently M: " o '^/3y and, since a^ = yp/p ^ y 0 2 / 3 , i t follows t h a t x/a^ « ( -C/c) and
K/a « •& , a p a r t from multiplying f a c t o r s of order imi-ty. The oharac-teristic time aiid length are therefore comparable Tri.-th -the mean time be"tvroen
c o l l i s i o n s of the molecules and the ixean free path respecti-vel;j'-,
Thus for t ~ 1 or l e s s a continuvtm theory such as t h a t forrnula-ted here can hardly be v a l i d and we should d i r e c t at-tention primarily'' to\7ards
-the case t >> 1 , where i t i s p l a u s i b l e t o use such a theory. For the sake of completeness some r e s u l t s f o r t = 0+ -will be given, howe-ver,
5» Splujtiqnsjfgr_jT^_^ j / i t ? .
The f u i c t i o n s B ( S ) and G ( S ) of eq. 39 can be r e l a t e d v l a "tlio z e r o -veloci-tS''-at--the-wall condition, e q . 3 6 , T/e find t h a t
C(s) = ( y - 1 ) v ' T V~1 +Vs''' B ( S ) ,
•vrfience the s o l u t i o n 39 can be v / r i t t e n ' .' '.
e (x J S ) = '.exp ( - X -VS*) + ( y - l ) Vs" VI '+ >s'exp(-sx(l + y S ) ' ^ ^ ) 3 B ( S ) . (^.2)
It should bc noted that eq. 2f2 is a valid solution for "the gas teraxirature in the half-plane x > 0 Yrhen u = 0 at x = 0 for an^ variation of 6
at this in-torface, only -the function B ( S ) changing in accordance vith the specified behaviour of 6 (O, t ) . For the present problen one readily infers from conditions 33, that
- (
B ( S ) = - ( T o/s) 1 + Q + (y- 1) s + Q( y - 1) V F /I + -fP (43) Using the inversion -theorem for Laplace transforms i t follov/s tliat
J..^ / exp ( t s - X Vs^) ds L i + Q + ( y - l ) s + n ( y » i ) - / i ' VTTys" 2 " i 'x - • Q + ( y . i ) s + n ( y „ i ) V i ' / f T y s ' ^ ^ T 1 + Q + ( y - 1 ) s + Q( y - 1) / s ' / i + y s Vs + JL f ( y - 1) ^ 1 + y s ' o:sp{ t s - sLl_+^ y si*"- x j . ds 2 ^ i 1 .u n X r ,, _ 1^= ^ nl'w _ -1^ V:S' Vi a . ' v ^ Vi' '
ihh)
e 9 • • •L being the tosual in-v-ersion contour.
The f i r s t and second i n t e g r a l s i n eq. 2f4 Tïill be denoted hy I^ and 12 rospecti-ve3y and each one t r e a t e d separa-tely belor/.
5 . 1 . Eveiluation of I ^
Tlie s i n g u l a r i t i e s of the integrand in I^ are branch p o i n t s a t s = 0 and - l / y . I t coja be v e r i f i e d t h a t L i s equivalent to a dunbcll contour proceeding p a r a l l e l t o the Re s axis j u s t below and j u s t above the branch cut betv/een s = 0 and - l / y and e n c i r c l i n g tlie branch ipoints a t
e i t h e r end. On the s t r a i g h t l i n e p a r t s of t h i s contour vro put s = y exp(± ±7r), y r e a l and p o s i t i v e , -taking upper and lov/er signs on the upper and lower p a t h s . Then Vs = •/j?-'exp (± ITT/z) accordingly and VT + y s = "/"I •-Vy* on e i t h e r pa-th. On the c i r c l e s surrounding the branch p o i n t s we put
10
-s =éexp(i6) and -s = - 1/9 + e e x p ( i Ö ) re-spectively and then allow e to approach zero. The contribution from the circle around s = -1/V vanishes in the limit and that from the circle roiaid s = O i s ( l + Q ) ' ' , as may bo easily verified. Rearranging the 'straight line' in-tegrations it folla/s -that
(45)
\7hen t >> 1 the principal contribution to -the integral here comes from the region near y = 0, Accordingly Tro can expand the in-te grand in ascending loovTors of y, -the f i r s t -term in each of the tvm integrals in eq, 45 being as follows
\ - - I I Q - •^k' - J - i - ( - ^ y V " ' ^ ' ^ -f-^ r ] oos(xV-y>-yt^
(46)
Replacing the upper limit l/y by oo, I can be written in tenns of T/ell known functions to gi-ve
(1 + Q ) I ~ 1 - erf(V2A'') _ ^ ^ . ^M^dMl. ^ (1,7)*
We must now in-vestigate -the errors in the result 47. First of all, in replacing 1/V by oo in the limits of the integrals in eq, 2).6 we imply errors cf tlie order of
1 ƒ " 3in(xV-y>-y^ ^ . ^ ^ ; ^ - ^ Ü 1 ƒ " cos(x/y)e-y*^4?
Vy ' > ^
on -the right hand side of eq, 47. By a mean value -theorem we can v/rite
1 J ~ s i n ( x / y > - y * ^ = ^ C - l | V y ^ • ^ = ^ ( . x ^ E,(t/y)
Vy Vy
erf(a) i s the error function, = (2/Vn^') / e~^ dy.
a
2
vAiere a i s a s u i t a b l e meaai value of Vy and E^(t/yJ i s the e^qjonential i n t e g r a l defined i n Bateman (1953 p.143 oq. 9 . 7 ( 1 ) ) . ^^hjen t/V » 1 , E j ( t / y ) ~ ( y / t ) o x p ( - t / y ) , (vide Bateman, 1953 p.14tj- eq. 9 . 7 . ( 7 ) ) . LikcTd.se vro can virrite
cos (x f y ^ e ' ^ * ^ , = cos(.xb) j e ' ^ * djr = cos(xb}_ , . e r f ( / t / y ) ^
y Ty '' 1/y /7 frt* w//y.
where b i s a s u i t a b l e mean value of v ^ . When t / y » 1 , erf(''t/'y) ~ 1 - V y / 77t exp ( - t / y ) . I t follows -therefore t h a t the e r r o r inipliod i n n e g l e c t i n g -the difference be-tvTcen 1/y and » i s of order
We no-ticc t h a t even when x approaches t quite c l o s e l y i n en, 47 the
e r r o r s hcic a r o a t most of order l / V t t i n e s -this r e s u l t , and aro correspondingly l e s s s i g n i f i c a n t as x •* 0.
The -terms neglected i n expanding the p a r t of the integrand i n braces i n oq, li5 give r i s e t o e r r o r s 1 + 0(y) i n the i n t e g r a l s i n oq, 1^6. I t can be seen t h a t the r e s u l t i n g e r r o r terms are obtained as second ojid tliird
dcri"vatives of the f i r s t i n t e g r a l i n eq, l{£ (ignoring constant multiplying f a c t o r s ) T/i-th r e s p e c t to x, i . e . successive d e r i v a t i v e s of erf(x/2Vt) TTith respect t o x as a not imre as enable ostiriiate. These d e r i v a t i v e s of
the e r r o r function contain a f a c t o r (l/Vïrt)exp(-xV4t) times e i t h e r x / t , or l / t and x^/t^. Provided x i s not of ordjer t the e r r o r s arc small con^jared VD.tli the terms w r i t t e n i n eq. 4 7 .
Pro-vided x i s not of order t , then, eq. It-l i s a reasonable reioresentation of the i n t e g r a l I , . ^^Tien x i s of order t or ^^^^^.ter i t i s c l e a r -tliat 11
c o n t r i b u t e s a small amount only to the o v e r a l l value of 6 , by ix^ason of the exponential terms. We s h a l l see now t h a t most of the conti-ibution to 9 i n -tlie region x ~ t comes from the second i n t e g r a l I when t i s l a r g e . We remark -that the complete s o l u t i o n of the ' c l a s s i c a l ' h e a t Cü'i.duction problem ( f o r which jjressure i s assunxjd constant throughout) i s given exactly by the f i r s t -two terms on the rij^it hand side of eq. 4 7 , tlie
s o l u t i o n i n t h a t case being v a l i d f o r a l l x and t . The intc^;ral I gives s o l u t i o n s of a purely diffusi-ve n a t u r e , as seeijis reasonable flcaii -the
presence of the e x p ( - x Vs) f a c t o r -there, 1.
12
-5.2. Evaluation of 12
To complete the solution for 6(xjt) it is novT necessary to examine the second integral of eq. 2i4, nanioly I2 . Vfe have just seen hov/ I1 leads to diffusion-type solutions and e^-jmiination of 12 may lead us to suspect -that -this integral will produce a combination of diffusion -type and wave-like solutions^ from the presence of the exponential factor
exp(ts - s(l + ys)"'^x), which has a character somewhere betr/een tliese -two. In fact just -this exponential -term arises in the s-tudy of pijrely viscous phenomena, mentioned previously as having been examined by
Lagcrstrom, Cole and Trilling. It is the complete transform solution of the Diitxc velocity pulse problem in a fluid for which n ;^ 0, X = 0 and, when multiplied by s~^, gi-ves the solution for a imit s-tep function of velocity ap;plied a t t = 0, x = 0. The above named authors have found solutions valid for large and small times by a subtle choice of contoiJir, folloT/cd by some lengthy sifting of various contributions to the whole in-tegral from different pai'ts of the contour in order to extract the most significant terms. Later Haniji (1957) treated the Dirac pulse problem
at groat length, finding solutions in series, as real integral representations and as asymptotic series, covering various ranges of x and t. Ilorrison (1957) also discovered the real integral repro s-^n tat ions for the inrpiolse solution, during the course of his investigations of wave propagation in
-vlsco-elastio materials, by using certain -theorems on Laplace transforms. The present problem is more difficult than any of these, hovra-ver, by reason of the complicated algebraic factor which multiplies the e3q?onential term in I2 .
It is clear that the two integrals I^ and I2 express "the "combined" nature of our problem quite well. The sudden changes of tempera-ture occurring first at the interface be-t\-roon gas ajid solid are boi:ind to
produce changes of pressure, density and -velocity in the gas, and one v/ould expect such changes to propagate out into the gas as some kind of wa-ve motion. Such wave motion is necessarily going to be of a somewhat complicated nature, since it is an essential part of the whole problem •that the dispersive and absorpti-ve mechanisms of conduction and momen-tum diffusion shall bo present. These T/ill act to change the form of the wa-ve motion, and these changes vrxll themselves react back on the diffusi-ve and convective processes which are responsible for the changing energy balance be-tvTcen solid and gas. Thus ITG may say -that the algebraic factor in I2 represents -the type of "input" to the wave motion in the gas as a result of diffusion Tjhilst that in I, represents the "input" to the diffusion processes as a result of the primary heat conduction processes plus -the feedback from the v/ave motion. It is characteristic of the
(J- = 3/4 case that these two types of process separate in the v/ay found in eq. 39. I't is clear from the form of the auxiliary equation 37 tliat such separation will tiot occur so obviously for other Prandtl numbers, and indeed -the solution 4I for o" = 0 provides a specific example,
To return to the problem in hand, namely the evaluation of 1^,
we shall concentrate as before on solutions valid for large timec;, folloT/ing very closely -the mo-thods used by Hanin. First of all xre
examine the region around x = t by defining
6 = X - t (48)
and malting the substitution (1 + yo)^ = f^. Then
I ^ l X . - , l J j ^ f i£L„exE.Jik:}lL'lldJIlj=lZ!^^
^ '^ Gy y (1+Q) +(y-i)(w^-i) + Q(y-1) yV (w^-i"jw Ycoli (49)
The con-tour Cu i s asymptotic to oo exp( ± 3?V'4) a t i t s ends and crosses the r e a l a x i s somcv;here t o -tlie r i g h t of w = -1. Yfe s h a l l imagine t h a t t i n e q . 49 i s l a r g e . 6 Td.ll be assumed small and vje s h a l l see l a t e r j u s t Tdiat -this must imply about -the a c t u a l allowable raagni-tude of ^ , Using the method of s t e e p e s t descents, i t i s now necessary t o f i n d , f i r s t
a s u i t a b l e saddle p o i n t f o r the function
f(co) = (co^- 1)(1 - 1 / c o ) , (50) and second t o ensure t h a t the s-teepest pa-th through t h i s p o i n t can be
reconciled vd-th -the contour C^^. The condition df/d'^ ~ ^i which defines the saddle p o i n t s of f(w) i s s a t i s f i e d by s e t t i n g w = 1 ( i . e . df/dw = 2w — 1 - l/w^ = O) and the s t e e p e s t jjath of descent from the c o l a t w = 1 proceeds from 1/2 - ioo , throu^jji w = 1 , t o 1/2 + i»» , w = 1 happens to be a singular p o i n t of the integrand i n eq. 4 9 , hor/e-yer, so
-that c l e a r l y the s t e e p e s t path for -the exjDonential function cannot be reconciled d i r e c t l y Tdth 0 . By indenting -the s t e e p e s t patli so as t o pass to the r i g l i t of w = 1 around an arc of a small c i r c l e given by u = 1 + e e2qp(ie) we may s t i l l make use of i t though, and i t quj.ckly follo\7s
t h a t the c o n t r i b u t i o n to I^ made by i n t e g r a t i o n around t h i s a r c approaches zero as e -. 0. Consequently TTC can now proceed i n the u r u a l way by Tnriting
f(a)) = - <f>V2, (51) thereby defining 0 , the r e a l v a r i a b l e of i n t e g r a t i o n on the s t e e p e s t p a t h .
I t follovra from 50 and 51 t h a t w expixissed as a s e r i e s i n 0 begins
u = 1 ± i 0 / 2 (52)
and that to a first order
1 4
-The upper and lavrov signs i n 52 and 53 are to be taken on the upper ( « = 1 t o - g ^ + i o o ) and lower (•g--i'=° t o 1) halves of the s-teepest p a t h r e s p e c t i v e l y ,
Taking the f i r s t -two -terms i n -the expansion of the integrand i n e q , 49 i n terms of ^ Tre ha-ve, a f t e r soEie manipiolation
- 4 y - ^ 4 ^ ^ . I ^ ~ ~ / e x p ( - t ^y2 y) cos (S0/y + V 4 ) ^ " ^ d?^ •
•"^C-fïo} • i ƒ ^ ^ (-*^'/2 y) cos (60/y ) d 0 , (54)
• oThe derivation of result 54 follor/s the standard procedure of the s-teepest descents method, namely series expansion of -the function of w^ ^ v^ich multiplies the exponential exp( t f(a))/y ) , in ascending por/ers of 9^, except -that we ha-ve written the term exp(-( 6/y)((o-l/u))as approxima-tely eqxial to exp(+ i06/y), taking signs appropriate to -the particular half of the s-teepest pa-th being considered. This approximation gi-ves rise to the cosine f-unctions in 54. The first integral can be reduced to a recognizable form on substituting (p = Vy 7'V' y, namely,
( £f 1 r ^"^"^^003 I" (V^v^y -.- V4 ] y"* ay ,
oBateman (1953, p.120 e q . 8 . 5 ( 4 ) ) showing t h a t t h i s i s r e l a t e d t o Weber's pajrabolic c y l i n d e r function of order —g- (T/rit-ten as D i ) . The second integreüL i n 54 i s a w e l l knoTm one and wo can Tra-ite
( 5 Z^'i)
(55)^yd+Q.)
^ 2 -( y / t ) ! 3 - % y t ^^ Q ( V - 1 ) e-«5'2ytA careful in-vestigation of the errors in 54 or 55 as approximations to I2 for t >>1 indicates that vio must restrict 6/Vy't' to be < 0(l) to prevent them becoming comparable Tdthgthe terms re-tained there, "i^en this is done the next term in 55 is o(t'"^, Oiving to the complicated na-tui^ of the integral it is impossible to give any general term for an asymptotic expansion, even in the present relatively manageable region of X and t. We observe, incidentally, that when 6= 0, D 1 is 0(1), (-the exact value -vd-ll be given later on in Section 5.3.).
The result 55 fails as a general approximation, valid when x is small for example, because of the behaviour of the exp |_-( 6/y)( w - l/w J •term in -this region, 6 becoming large like -the time t.
Howe-ver, if one assumes -that 6 _is large end negative, as when x is small, it can be shovoi (using^-the results of Miller (1955)) that "the first term in 55 behaves like S-a" and -thqit the first error -term from the exponential just IiTentioned behaves like 6^'*. This suggests that at small values of x (i.e. near "the interface), there is a part of the disturbance which is of a wa-ye-like character and that it may be of a coniparable order of magnitude to the third term of eq. 47. Physically -tliis state of affairs a:p]_X!ars
highly plausible and accordingly we will attempt to e-valuate the contribution which -the in-tegral I makes to 6 in the regions of x near to -the interface.
2
To do tliis i t i s observed t h a t the exponential terra i n -tlie in-fcegral 49 can be re-vait-ten as
expfXo)^ - 1 ) ( 1 - a / a ) ) ( t / y ) j (56)
MJherc
a = x / t , (57)
We now sock s o l u t i o n s f o r 12 Tdiich are v a l i d when t>> 1 and ct i s small, Using steeiD^st d e s c e n t s , the c o l f o r the function
g(co) = ( a ) 2 - l ) ( l - a / o ) ) (58)
is found as a solution of
g'( w) = 20) - a - a/u" = 0 , (59)
a i s r e a l and p o s i t i v e and i t follor/s t h a t there must be one r e a l and -tvTO
complex roo-ts of eq. 59. Of these \/e choose the r e a l r o o t , noting -tliat when a i s siiiall t h i s r o o t i s approxima.tely
CO
o
(é ['*3¥*••••]• <«°)
The s t e e p e s t path of descent passes through w _ o) and i s asyniptotic•to ( w - "/on^ ) - i o ^ a t i t s ends. However, -the a l g e b r a i c faustor i n 0
eq. 49 contains a branch p o i n t a t w = 1 and the o r i g i n a l contoijr C ^ cannot be reduced to t h i s s t e e p e s t path d i r e c t l y . I n s t e a d we s h a l l use the contoior i l l u s t r a t e d i n F i g . 1 iThich c o n s i s t s of the upper and loiTer halves of the s t e e p e s t path connected by a loop aroiaid the branch c u t from u = 1 .
16
-The integration around -the small circle of radius e contributes an amount to I which is proportional to ve ^ and consequently yields zero in the limit as e ., 0. Writing Re w = v, the parts AB, CD of -the contour contribute an amount
2 ( y - i ) V y '
- V^
iJ expC-(l-v^)(l-o/v)(t/y)1 vfdv
)p + Q^(y-i)^yv^i-^^
'i^:^
(61) The index of the exponential term hero i s zero a t the xipper l i m i t V = 1 and i s negative e-veryv-jhere e l s e w i t h i n the range of i n t e g r a t i o n .
Ti/hence i t follovirs t h a t vdien t>> 1 -the p r i n c i p a l c o n t r i b u t i o n t o the i n t e g r a l w i l l come from the region near v = 1 . Changing the v a r i a b l e from v t o y •via the r e l a t i o n ( l - v ^ ) ( l - q / v ) = y , the integrand can now be expanded as
a povTer s e r i e s i n y , -the most importajit term being (l-(/ )(l-a/co^)
/y'(i + Q)
^TZa'
1
Ir
,-yVy
y ^ dyV5Frt--'xy
erf
(1 -
^l
)(i -«/
%){Wy)
(62)The errors in writing 62 for the integral 61 are very small conipared A"dth -the resiilt 62 pro-vided a is less -than about l/2 and t is large. 0\"dng to -the complicated nature of the integral it is not practicable to give a general result for the error terms, but they are roughly of ordjer t~^ exp l-(1 - w )(1 - ^/w )(t/>) I , Since the argument of the error function in 62 is very large if a is small -VTO may reasonably approximate
to -the eacpression there by writing it as
1
v~^xt~-xr
(63)To c a n y out -the integration along the steepest path part of the contour in Fig. 1 -vre define the real variable of integration y as follor/s,
g(a)) - g(w^) = - y^ ,
The usiial procedure for evaluating integrals by the me-thod of steepest descents -then leads to -the major term arising from -this part of the contour and -this is found to be
2 Vy' ( y - 1) A cj^ ( y/rrt) exp (g(<JQ)t/y)
y ( i + Q) - ( y - i ) ( i -'^^)+ Q(y- i)Vy'co^Vi _a;^"^
where A i s Tnritten for l / ( l + a/w^). Since g(u ) - - 1 when a , and hence cj , i s small, t h i s term i s very much l e s s than the r e s u l t 63 and T\fe conclude t h a t , provided t >>1 oxid a i s reasonably. siiitH coLipared \7itli u n i t y , a p l a u s i b l e estimate of the i n t e g r a l I i s
This r e s u l t confirms -the view, expressed e a r l i e r , -that T/ave-lilce dis-turbances e x i s t i n the regions of x near to tho i n t e r f a c e .
As i n Hanin' s paper, the form of the exponential function i n oq. 56 can be used t o find approximations for the case t >>1 and a l a r g o a l s o ,
I n -that event-the s o l u t i o n of eq. 59 i n d i c a t e s a saddle p o i n t a t approximately CJ = ( a / 2 ) , and "this T d l l c e r t a i n l y l i e to the r i g h t of the branch
p o i n t a t 0) = 1 . Consequently C and the s t e e p e s t pa-th are diroctl^'-reconcilable and i t can -then be sho\-/n t h a t -the major c o n t r i b u t i o n t o 8 from I i s roughly
2
1+Q Vy
which i s a very small q u a n t i t y , ( i n dcri-ving 65 V/G have neglected uni-ty i n comparison Tdth w^ . ^ OL^ /k) .
5 . 3 . The Tenrperaturc a t Large_ Times '"'' " ' ••'• • >' '"
C o l l e c t i n g the r e s t i l t s from the l a s t three s e c t i o n s enables us t o bvdld up a reasonable p i c t u r e of the beliaviour of the gas tcrTperature a t times large compared Tdth -tlic moan time bc-tv/oen molecular c o l l i s i o n s .
Thus, i n the region \7here x i s small compared Td-th t
e(x,t) ~ . f^ [ i . erf(V2Vt) - ^ ^ Ü . - ^ ^ A t i
+ X.-J™ ! (66) fTrCt-xT-^
18
-e(x,t) ^ - ? ^ ^ . (y- A e:^(-5V4yt)D^(6/Vyt-)
V^' (yt)^
exp(-5V2n)
^ -, , - (67)
V2 wyt
Por the l a t t e r case we have used the r e s u l t s 2^4 and 55 and i t i s r e c a l l e d t h a t 6 at X «• t . When x i s g r e a t e r -than t by an appreciable amount -the value of 6 1ms been shovm t o be p r a c t i c a l l y zero,
Tlie teniperature a t -the inteyji'DXje x = 0 follows from eq. 66, namely,
e ( o , t ) ~ - ^ [^^Lu:Jl ( ^ ) 4 ] ^ (68)
and, remembering t h a t -the gas tempera-ture T = 6 + T , i t can bo seen -that the i n t e r f a c e -temperature i s i n c r e a s i n g with time, i f T^ i s p o s i t i v e ( i , e . gas h o t t e r -than s o l i d ) . Tho c l a s s i c a l cons-tant pressure s o l u t i o n for 6 ( 0 , t ) indica-tes -that i t jumps abruptly to the valiie - Q T ^ ( 1 + Q ) and remains constant for a l l l a t e r times. Thus i n -the prac-tical case the c l a s s i c a l s o l u t i o n i s approached asymptotically. This statement i s a l s o •true of -the -vrfiole s o l u t i o n f o r 6 i n -the region near the w a l l , since as t increases the l a s t -two terms i n eq, 66 become small ccai^jared wi-tli the f i r s t •two (v*iich represent the c l a s s i c a l solution) . As distance from the TTall increases a t gi-^ren -time, howe-ver, -the s o l u t i o n 66 i n d i c a t e s tliat deviations from -the c l a s s i c a l s o l u t i o n increase and i t seems p l a u s i b l e to suggpst t h a t such deviations tend t o become of a predorainaiitly wa-ve-like c h a r a c t e r , The behaviour of -the l a s t -two terms i n 66 as x increases i s such as t o caxise the gas tempera-ture t o f a l l below the c l a s s i c a l v a l u e . Turning now t o -the regions T^ere x and t arc of comparable magiitude we f i n d (from eq. 67) some notable de-viations from -the c l a s s i c a l valve of 6 ( x , t ) . This lat-tcr s o l u t i o n Trould i n d i c a t e t h a t 6 has f a l l e n to an ^almost./„
n e g l i g i b l e s i z e when x = t , f o r exaiif)le, because 1 - e r f v't/2' ~ e**^ /^^^^/S* when t i ü l^irgö, Eq, 67, hot/e-ver, shows t h a t
e(t,t) ~ .^. ( , . , ) f - ^ r - - fe} -^ ]m
^^^ (^2vr(2vt)^ m •<• Q; /2W^? ->
3 i
( N B . D t (0) = It^)/2'* w^), •vrfiich, although small because t must be l a r g e , i s c e r t a i n l y of a g r e a t e r order of magni-tude than -the c l a s s i c a l s o l u t i o n ,
A ske-tch of the coniple-te tenperaturc dis-tribution i s given i n P i g , 2 •the f u l l l i n e curves being c a l c u l a t e d from equations 66 and 67, w h i l s t the dot-ted l i n e s isspresent a p l a u s i b l e estimate of the boha-viour of the temperature i n -the regions Triiere -these asynptotic s o l u t i o n s f a i l . The c l a s s i c a l
s o l u t i o n i s shovm f o r ooinparison, and i t can be seen how -the de"viations from •this s o l u t i o n becone more marked as x i n c r e a s e s . P i g . 3 i s a sketch
of the wave front for -two values of t (50 and 100), and indica-tes how its amplitude diminishes and how it becomes more diffuse as time inci'eases, These -two effects arise from -the dissipative actions of -viscosi"ty and heat conduction. In an ac-tual case the non-linear terms in tlie equations describing the motion (which ha-ve boon neglected in our linearised
t]rea-tnient) would act to flatten the wa-ve front even fur-ther* Bo-th Pig. 2 and Pig, 3 ha-ve been drawn for a -valxie of Q = 100, -which is roughly the magni-tude of -this quantity for an air to pyrex-glass contact. This is
the se-fc-up generally encountered in -the use of thin-film platinum resistance -thermome-fccrs in shock-tube work. The variation of interface temperature from -the classical valxae is far too small to appear on Pig. 2 vd^th this particular value of Q('vide eq. 68), so that Pig, 4 shows a sketch of
T ( = 6 + %o) ^'^ ^'^ in-terface plotted against time. A rather more accura-te es-timate of this value is made in Section 5.5. below and Pig, 4 is a plot of eq, 85 appearing -there. It can be seen that for t > 100 -the differences be-tween actual and classical values of T are insignificant for all practical purposes. For conditions around N,T,P, the mean collision time is of
order 10"*^*'seos., so -that no difference from the classical solution would be observed for -times greater thaji about 1/I00th of a microsecond,
which implies -that the practical effects of compressibility in heat •transfer at the in-terface cannot be resolved experimentally.
5«4. The Pressure and Velocii
Further comment on the significance of the results obtained abo-ve Tdll be given in "the final section: Tve proceed now to consider the pressure and veloci-ty pertxirbations which must arise in -the gas, Tho
linearised energy equation (13) in dimensionless form is
*2( a£ = „ a r 9 i - Ë!e = p a t ' ^ ^ O O P ( ^ 9 t g ^
%
U t - r r (70)
•which gi-ves <x> s P = p^ Cp j s ê - e" I (71)i n -the transform p l a n e , provided p r e p r e s e n t s -the transform of p - p^ . Eq, 42 then shows t h a t
p = P^ C B(s) ( y - 1) / a '^ , ! j l £ ^ / - ^ exp - s x / V l + y s ' ,
P Vi + y s ' L J (72)
B ( S ) ha-ving been given i n eq. 4 3 . I t i s observed t h a t p can bo w r i t t e n as • I n the event-that "the gas i s i n i t i a l l y colder than -the s o l i d ( T „ < O) •the wave f r o n t i s one through -vdiich temperature i n c r e a s e s . I n t h a t event •the n o n - l i n e a r convecti^ve e f f e c t s w i l l co\aiteract -the dispei^i-ve e f f e c t s of viscosi-ty and heat conduction and the wave f r o n t T d l l tend t o remain s t e e p , i , e , i t •will be a shock -wave •whose streng-th T d l l depend on T^o and Q,
20
-r. / 1 + ( . y - i ) s ) 5
•where we w r i t e 9 for the second -term i n eq, 4 2 . Consequently ^
p - . p = - P C Q T . l '
•v\4iore I' is an integral exactly like I in eq. 44» except that its in-tegrand is multiplied by (I + ( y- l)s)(l + y s ) " , i.e. an integral like eq. 49 whose integrand is multiplied by ( y- 1)/V + lA'^^ •
It follows that, using the steepest descents approach when x is of order t (i.e. for Ö "small"), -the first -two terms of I^ will be identical Tdth the first -two te-rms of I and Tve can To-ite directly
P - Poo "^ ^00 % ® ( ^ ' * ) ("^^^
•vrtiere e ( x , t ) i s gi-vcn by eq. 67.
By -very s i m i l a r arguments Tve can i n f e r t h a t a f i r s t order estimate o^ P -• P» f otr X small i s
The v e l o c i t y induced by the heat conduction T)rocesses can be found from -tho non-dimensional version of eq. 1 1 , namely
' -co 1^ = p ^ p ^ y - - » ^ - 1^ • (75)
CO CO g x 00 p 2 2 o x
I t follows t h a t -the trans.form of the -velocity ü i s gi-ven by
p a Ü = P G ( y - 1 ) ( ^ ' ) ^ - s p dx. (76)
c o c o c o p ^ o J ' • ' • '
Making use of the pre-vious results for 6 and p (eqs. 42 and 72 respectively)
•we find that
P a^ Ü = -P^ C ( y - 1) V"3 B(s) [ exp(-x/s) - exp(-sV>^1 + ys) J
'^ "" (77)
The f i r s t i n t e g r a l which must be evaluated t o find u ( x , t ) i s vcrj'' l i k e I , i n f a c t i t i s I, -with -the integrand multiplied by -Vs, and can be trea-fced i n a s i m i l a r fashion. I t i s found to contribu-te an araoimt
,^ / ^ \ T Q -X V 4 t p G ( y - 1) oo/*' e ^^ t o the whole value of P a u
Inspection of the second term i n eq. 77 shovvs i t to be sii":il.ar to the Ij f o m of i n t e g r a l , -the only difference being t h a t i t s integrand w i l l be t h a t of I^ divided by ./l ï ^ s ^ and the pro-vious remarks made about Ig follov/ here t o o . V/honce we can virri-te a t once,
T O r - x 2 / 4 t J, -)
. P « a , p - P C ( y - 1 ) ^ ^ r ^ _ ^ . I (78) when X i s siiiall, and -when x i s comparable Td-th t
PcJ^o.^ =^ P^ Cp Q ( x , t ) - P - P „ , (79)
6 ( x , t ) being given by eq. 67.
I t i s of i n t e r e s t to observe -tliat the pressijre, veloci-ty and •tenperatiire p e r t u r b a t i o n s i n -tlie region aroimd the T/a-ve front a r e , t o the accuracy of the s o l u t i o n s prcson-ted h e r e , e x a c t l y -those for an infini-tesimal i s e n t r o p i c siniple wa-ve, (-vide e q s . 73 and 7 9 ) . Tliis i s not s u r p r i s i n g , s i n c e , i n the l i n e a r i s e d s o l u t i o n , -the i r r e v e r s i b l e e f f e c t s of h e a t conduction and v i s c o s i t y are neglected and, a t t h e large tiiiies for which our s o l u t i o n s are v a l i d , the ac-tual quan-tities of heat conduc-ted i n t o and out of the wave front are si;iall. This l a t t e r statement draws some support from the sketch of 6 ( x , t ) i n P i g . 2 , which shows liov/ f l a t -the d i s t r i b u t i o n of tempcratixre i s i n t h i s region. IThen x i s small, hov/evor, the i s e n t r o p i c c h a r a c t e r of tlie dis-turbances v a n i s h e s , as i s e-vident from e q s . 74 and 7 8 .
Eq. 75 shov/s -that the pressure p e r t u r b a t i o n i s an expansion across •the region x ~ t f olloived by a gradual recor>Tpression as -tlie i n t e r f a c e i s approached. The v e l o c i t y disturbance i s c o n s i s t e n t wi-th -this pressure d i s t r i b u t i o n (see e q s . 78 and 79) > and i t i s c l e a r t h a t as tiiix) increases -the system approaches the c l a s s i c a l heat conduction conditions of ccnstant presstire and zero •velocity.
5 . 5 . Intcrfa.ce Tenpora.-ture_ and Conditions a t Zero Time
Before going on t o consider the cr = 0 case vre s h a l l b r i e f l y examine -the temxjcrature a t the i n t e r f a c e i n a l i t t l e g r e a t e r d e t a i l and a l s o -tlie conditions a t time t = 0 +,
uTien X = 0 , tho transform solution for -the temperature reduces t o
ê(o;s) = - M j ^ j y ^ J L S ï L t l y ^ . . — - . , ^
^ 1 + Q + (y - l ) s + Q(y - 1) Vi'^1 + Vs
22
-(see e q s . 2f2 and 4 3 ) . ''Jnience i t f e l l a h s tliat tlie T/all teniperature can be Tvritten as
iCo.t) = !^ ƒ ±lSl2±,ïll ^
L 1 + Q + ( y - 1) s + Q( y - 1) Vs"' Vl"+ y s ^
(81)
L is the usual inversion contour, but it can be deformed into a dumb-bell contour surrounding the branch points at s = 0 and - l/y. Y/hen t » i -the in-tegrand in 81 can be expanded in ascending powers of s, 'tliG first -three significant terms giving
0 0 . . . ( 8 2 )
A = ^^rfq^ (83)
•yrfiei^
(i.Q)^ 20.0) (trof^
(84)The i n t e g r a l s i n 82 are incoinple-te gamma functions which, however, d i f f e r a n e g l i g i b l e amoimt from the coinplete -values when t » 1 (-vide Bateman, 1953 p . 1 3 5 ) . Consequently we can wri-te
T ( 0 , t ) . _ I ^ r 1 - ^ . B -j ^ (35) 1 + Q L VTTt aVrr't'z J
« 5 /
-the next term being of order t '^ . I t can be seen how ccanplioa-ted the c o e f f i c i e n t s arc becoming, even i n suq^L a simple case as the p r e s e n t one.
Al-üiough the large time condition has been s-tudied exclusively f o r the reasons s t a t e d i n Section 4» i t i s of i n t e r e s t to look b r i e f l y a t
the small time p r e d i c t i o n s of continuum theory. Since -the major departures from ambient conditions w i l l a r i s e when x i s small a l s o f o r small times, we s h a l l content ourselves Tdth -the i n t e r f a c e values of p ajid 6 a t time
t = 0+, These can e a s i l y be found from e q s . 72 and k2 by l e t t i n g s -• co , •v^enoe
p(o,o) = - p ( c A ) ( y - i ) T ^ . ^' ^ , , (86)
^ 1 + Q vy' T ^ Q V > ? 6 ( 0 , 0 ) = - — - . (87) 1 + Q A 'These r e s u l t s show -that, i n i t i a l l y , -tlie density has not changed ( N , B , ( C / y ) ( y - 1) = R, -the gas cons tan-t, whence tlie constancy of densi-ty follovTs xrom -the f i r s t of equations l ) and -that the whole jprocess begins as i f i t i s t o be one taking place a t constant •volvime. This l a t t e r f a c t i s apparent on observing the d e f i n i t i o n s of Q (eq. 34) and '^ ( e q . 14) when i t cati be seen -that TTriting Q Vy"^ i s equivalent t o redefining ; ^ e
diffusi-vi-ty K i n terms of ^^Y» the s:pecific heat a t constant volume. Eq, 87 -tlien r e p r e s e n t s "tlie c l a s s i c a l h e a t conduction s o l u t i o n appropriate
t o t h i s type of p r o c e s s . The continuum solution thus r e p r e s e n t s a
t r a n s i t i o n bet,'jeen the -two processes of heat conduction a t constant volume and h e a t conduction a t constant p r e s s u r e ,
These remarks conclude ovir trea-traent of -the o" = ^ case and we sliall now examine -the case ^- 0 t o see how i t v a r i e s i n i t s behaviour from the r e s u l t s gi-ven above,
6 , Sol-u-tions f o r ^ = 0
Y^ien cr = 0 the s o l u t i o n 41 niust be used. The constants B and C can be e'valua-ted from conditions 33> 34 and 36 and -the s o l u t i o n f o r 6 ( x , t ) expressed i n -the forai
Q T^ f (v, + v j ^ l - v^- v j e ' -(v, - v j ^ d - v^ + v j e " i
e(x,t) =
2 v r i (v, + v j ^ „ ( v „ v )2jj^(v^+vJ2+(v^-v^)''^-t<}(l +yVs) (88)r^
S Y S •vriiere 1 > 2= -
rv{v, t vj'
2 * 1 + y s '2^ - s (89)Eq, 88 i s a -very vaiwieldy expression and even approxima-te e-valuation of -the in-tegrals seems impossible vdthout f i r s t attempting SOBTC kind of
•transformation which w i l l siniplify -the exponential terms t h e r e . Fortunately -the necessary transformations have been supplied by .Cole and Wu (1952),
but care must be -taken i n t h e i r a p p l i c a t i o n and accordingly some useful general observations about the i n t e g r a l s i n eq. 88 T d l l be made h e r e .
Closing the s t r a i g h t l i n e contour L to the r i g h t Tdth a s e n i - c i r c l e , whose r a d i u s R w i l l be allovred t o api:)roach i n f i n i t y , i t i s found t h a t -the jreal p a r t s of the exponential terms beha-ve as follows. The term i n
exp (ttg x) behaves l i k e exp(tR c o s 0 ) , ( N . B , s = R e x p ( i 0 ) ) , and hence c o n t r i b u t e s t o 6 ( x , t ) for a l l t > 0 , w h i l s t the term i n exp(a,x) behaves
24
-Referring t o -the d e f i n i t i o n s of the dimensionless co-ordinates x and t , i t can be seen t h a t the l i n e x = t/vV' indj.cates a_yeloci-tj'- equal i n magnitude t o the isothermal sound speed c^ = a„ /VV' . Across "this l i n e , -therefore, there i s a d i s c o n t i n u i t y i n the r e p r e s e n t a t i o n of the s o l u t i o n 6 ( x , t ) and i n f a c t Cole and ' U have shovm such l i n e s t o IM c h a r a c t e r i s t i c s of the "1^ = 0 system" of equations which leads to the r e s u l t 8 8 . Bearing i n mind -üie remarks made above, we now apnply C o l o ' s andV/u's transformations to 88. There arc two stages of transfoniTation, !Ehe f i r s t , common t o both -terms of 88 c o n s i s t s of Trriting
y^s = (1 + b w)(l - b / w) (90) where
b = Y y ' - ' T ,
whence eq, 88 becomes
•:. 0)^1 + QVy''[w2Vïlb7a)'+ VT+bco'] (l+bw ) ( l - b / u ) ^ (91)
e 5 E i l l ± ± ' d l L i i b / c o ^ ^ ^^ ^
. . . aj^+ 1 + QVy' [cj^ / l - b / w + ViT~b t ? ] y w^ (l-b/co ) Vï+bw''
e(x,t) = - 2 i ^ ^ o o
* 2^rigbVy-T, 2'JK'^
A possible contour C^ starts from w = - ico and proceeds -tov/ards to = ioo passing to the right of the singularities of the integrands.
These latter are branch points at («'s 0, b and - l/b and it can be verified -that the C^ aescribed above can be replaced by a contour which comes
from oj = oo exp(-i w"), loops around w = b and returns to >» exp(+ i^^^), This second form of C^ vdll shortly be found useful.
Each integral in eq. 91 is now tackled separately. Talking the first of these first, it should be noted that this is the CJ- plane
•version of the "exp(a x)" integral in the s-r)lane v/hich has been shomi to contribute to ö(x,t) for all t > 0. Since the second or ''exp(«,x)"
in-tegral is zero for x > t/^y' -the first integral in eq. 91 gives -the whole solution for this condition. '.Tc novT v/rite
OJ (^^ - i ) A ,
which -transforms the f i r s t i n t e g r a l of eq. 91 i n t o
e(x,t) _:
X > t/Vy'
QVy"' T,
TT i
y-1
.%2-y 1 +QVy' exp(92)
X
fy.
The contour G^ i n the ^ - p l a n e i s i l l u s t r a t e d i n F i g . 5 , the s i n g u l a r i t i e s of the integrand being branch p o i n t s a t S = ±Vy and - 1 vd-th a simple pole a t S = 0.
l"/hen X < t/Vy^ i t i s necessary t o consider the contribution, from •the second, or "exp(ajx)" i n t e g r a l i n eq. 9 1 . Y/riting
0) = - b / ( ^ - 1 ) (94)
it can be shown that -the integrand transforms into precisely -the form gi-ven in eq. 93. The contour of integration for •this second integral will be different from C^ , however. Making use of the second foiïn of Cy contour described above it can be shorm. that an appropriate S -^lajie contovir joins the points ABCODEF on Fig. 5, in the order v/ritten. The points A and F lie on the lov/er and upper halves of the branch cut be-tv/een ^ = V V and 1, just to -the right of the singularity at S = 1.
To find the solution for x <t/Vy' it is nov/ necessary to add -tiie integrials taken along C> and -the contour A to F, Since their integrands are
identical it is clear -that the parts of G> be-tr/een BC and ED are cancelled so -that integration should take place along the new contour C_^ which is illustrated in Fig. 5» (i.e. replace the symbol > by < in oq. 93).
The difference be-tv/een the contours which are necessary here and in the case treated by Cole and IVu is apparent. As will be shortly seen, and as may be inferred from their results, the difference is associa-ted TTi-th the presence of wa-ve-like phenomena in the region x < t/'^y* as well as when x > t/ V)?,
lar.fre
The part of C< to the left of ^ = 1 can be deformed into -tlio ImS axis, retaining -the indentation at ^ = 0, of course. This latter part of the contour is then a semi-circle and contributes an amount
•^ Ty(l + Q ) to e(x,t) for X < t/vV , Writing Im ^ = n and
{? +y)/(>7' + 1) ±
= f J Q V 7 = Q'
for bre"vi-ly, the remaining contribution to Q from C reduces to the real integrals
2 Q ' T
i j j + 1) + ^ u + i y j j „.!^ / XT? ^2 sin { ^h . f' '^ Ii (f + Q')'+ ^'(1 + " Q 7
(95)
26
-When t » 1 the p r i n c i p a l c o n t r i b u t i o n t o tlie in-tegrals 95 comes from -the region rj ~ 0. I n f a c t ^ = 0 and Re S = 0 are a saddle p o i n t and s-teepest path f o r the o r i g i n a l intcgiral, as can be e a s i l y v e r i f i e d . I n t h i s case f - •/y' and the e g r e s s i o n 95 i s approximately
Q T.
r r ^ Q . I ƒ e-*^>sin(vv->^)n-^d.
0. ^ ( y - l ) Q " T o o 2 f ' - t n V y , , r . N ^^^^ + — _ , £ e ^ '^cos( rix/Vy )dri ,
V-y (1+Q)^ ^ {
•In more familiar terms, the part of C^ to the left of ? = 1 contributes
to 6 (x,t). The errors in 97 are negligible vrtien t » 1 provided x is not too near t/-/y"',
The solution for x < t/Vy is completed by evaluating the integral 93 along -tiie contour ABEP, The circle surrounding ^ =Vy"' contributes zero to 6 in the limit as its radius approaches zero and vnriting Re ^ = 5, •the straight line parts of the contoior yield
. ^^%.r ^ ' - i V f g ( ^ ^ Q V y ' ^ ( g ^ - i ^ ^ ( y - s^Hi +gQV^')
Vy-gV l ( ? ' -1)(? + Qv-y7 + ( y - e)i^ +^oy?f
IT
exp
- 1 y y 2 3. 2
(98)
This i n t e g r a l can be trajisformed to an i n f i n i t e one by sxibsti^tuting
y ' + 1 whence, v/riting
[(y^+y)/(y'+ D.
^ = g J QV"y' = Q' for b r e v i t y , T/e have. ^ r y - ^ ^ T ^ r - r K ( « + Q') + y ^ 1 + f ^ ' ) ^ ^ ^ w •' ( g ••• 0 ^ Q ' ) ^ y t i + g Q ' ) ^
( y %
,/Vy^ yy2/.
i ) ( y % y ) ciy (9<Once again "this integral has a significant contribution mainl;>- near y = 0 when t >>1 and this, to a first order, is
. il - Ji
SJ^
2
vy" 1+Q
^'
expvy
yj^
dy .(y-
I)Q^
1The errors are small so long as x is not too near t/Ty' , for t » 1, The whole solution for x < t/Vy* is made up from the suiu of the expressions 97 and 100, namely
(100)
e
(x,t)
X <t//-i?
Ill'
1 + Q 1 - erf(x/2Vt' ) -^Lï^Jü. exEL-s'Z^l .^(y-iJ_
(101) To a first order of approximation then, this solution is identical with -the "small x" solution for o" = ^, as can be seen from eqs. 101 and. 66,
despite the api^arently very d.ifferent exact solutions (eqs. 4^;. and 88). It is worth noting that there does exist a certain similarity'- be-fcween the j3on-tours which ha-ve been used to obtain these results, have-vcr. Difference"be-tv/een the c = 0 and ^ cases would arise in higlier order
terms than those presented here, but these differences aro clearly'' of no great physical significance at large times,
It is perhaps a little surprising -that the agreement bet./een -üic -t\70 seemingly so different cases should be as good as has just been demonstrated here; in fact wo shall find that it is not qui-te so good when the x ~ t regions axo compared. The "small x" region is one in which diffusion effects predominate, hou^e-ver, and presTxnably Prandtl number is of less significance in these circumstances,
To examine 6(x,t) in the. region x ~ t va-ite X = t + 6
so -that the exponential j»erm in eq. 93 becomes
exp
( % L ) . Si(,.
Vy^
exp _ ^ / ^ - y N, 3/2y / \^ ^ ^ 1./
28
-When t » 1 and 5 i s small the method of s t e e p e s t descents can be used to find an appropria-te i n t e g r a t i o n ccantoi.ir T/liich has a r a p i d l y decreasing value of the f i r s t exponential f a c t o r i n i n expression 102 a l o M i t s lenc^fch, A saddle p o i n t f o r the function of ^ i n t h i s term i s ^ = »/ and the s t e e p e s t pa-th of descent l i e s be-tiTcen -this p o i n t and -the -points
S = Vy "' / 2 ± i CO , A small s e m i - c i r c u l a r indentation of -tliis pa-th i s necessary t o avoid the singulari-ty a t ^ = / • / ' which occurs i n eq. 93» but -this c o n t r i b u t e s zero -to -the f i n a l value of 6 ( x , t ) , I t can be v e r i f i e d -that the s t e e p e s t path Tdth -this indentation i s equivalent to G^ ,
Writing
S- ( S ^ - y ) ( S . V y ' ) ( S ^ . l ) - 1 = ^ <t>\ (103)
and -tliereby defining -the r e a l varJablo of i n t e g r a t i o n ^ on tho s t e e p e s t pa-fch, i t i s observed t h a t expanc?" r^n of S i n ascending powers of p begins
S = V/ ± i
J ^ ~ <t>
+ (104)
The uipper sign i n 104 r e f e r s t o -the upper h a l f of the path and -vice v e r s a , The f i r s t order estimate of eq. 93 Tdien x ~ t and t » 1 can new be writ-ten as
0(x,t) ~ - ^ ^ 2 ^ ( y - l j J 1 f " - t 9 i y / /V2'6^ ^mM'^
^^ Q V ? ^ J \ y(y-i)^ ^/7^
0 (105) P u t t i n g t<P /y = y / 2 •Klis becomese(x,t)~ - ^ . • ^ • M ^ . 4 / e - y > c o s r - ^ - - f ^ ^
t^ "^ i ^V^yTiTt^ ^ / Vy
(106)vdiich i s recognizable as the parabolic c y l i n d e r function form found i n Section 5 . 2 , In other words
e ( x , t ) ~ - ^ . h : ^ e x p ( - 6 V 4 ( y - i ) t ) D , (5/V"(y-iTt'), (107) 1+Q / 2 ' ^ ' t ^ " ^
a restilt v/hich should be compared vdth the first term of eq. 67 in
Section 5.3. They are seen to be identical if yt in eq. 67 is replaced ty (y-l)t.
When U = O -Uien, the form of the disturbance in the regions around the wave front is the sarae as in the more practical ct^se for which M
is re-tained, but the shape of the dis-turbaiice is much sharper at a gi-ven 6 and t. This is hardly surprising, since ixi the o" = 0 case the dispersi-ve effects of -viscosity are absent. It appears from the i-^sults 107 and 101, •Vivien compared v/ith the corresponding 0"= -4- solutions, that -tlie ^ = 0
approximation is indeed not as drastic as might be supposed in the first place, pro-vided t is large enough Also the effects of Prandtl nuniber
seem -to be of most importance in the v/ave-frcnt zone and of little sigiiificance near to tlie interface, certainly as tear as temperat-ure is concerned,
Dis-turbances
The disturbance to the initially imiform pressure and veloci-tj'-fields can be found in a similar manner to the cr= ^ case, althougli by reason of the conditionju= 0 here, -the velocity problem is in fact somewhat siinpler to solve. We will not gi-ve the details here to avoid •wearisome repetition, but merely quote the results.
Thus it is found that, v/hen x < t/Vy and x is not too nearly equal to t/-Vy'
T ^ i l s t v^en x ~ t
p - a , - p^ Cp 6 ( x , t ) , (109)
9 ( x , t ) being gi-ven by e q. 107.
Eq. 108 i s i d e n t i c a l wi-th eq. 74 Mid 109 i s the equivalent of the r e s u l t 7 3 , t o v/hich i t reduces i f (y - l ) t i s r e p l a c e d by y t .
Like-vd.se, -the v e l o c i t y u i s e x a c t l y the same as e q . 78 when x < t / ' V / ' and a r e s u l t e q u i v a l e n t t o 79 i s obtained v/hen x ~ t . To the order of
accuracy of these r e s u l t s , the s i m i l a r i t y be-tv/een the e ( x , t ) values f o r O" = 0 and 5 i s r e t a i n e d f o r the o t h e r flov/ v a r i a b l e s t h e r e f o r e .
I t can a l s o be shov/n t h a t the t = 0+ values of 6 and p are i d e n t i c a l i n the -tv/o c a s e s , so t h a t the t r a n s i t i o n from constant volume t o constant pressixre heat t r a n s f e r occurs v/hen c = 0 , t o o . Some idea Of the e x t e n t
of the difference be-tvveen the t\TO s o l u t i o n s a t the i n t e r f a c e f o r large t can be gained by cori-rparing e q . 85 with -the corresponding r e s u l t for o" = 0. I n -the l a t - t e r case T ( 0 , t ) i s gi-ven by an equation e x a c t l y l i k e 85 i n which the c o e f f i c i e n t B i s nov/
30
-For -typical values of y and Q of 1-,4 and 100, B above equals 1,94 v/hilst B in eq, 85 equals 0.49, but, since the result is only valid when t » 1 , the effect of this difference on T(0,t) is very sraall.
The difference be-tween the o" = 0 s::id o" = ^ oases has been found to be -very small at large times, the principal effects arising in the v/a-ve front region v/here viscosi-ty acts to disperse the disturbance more rapidly than can be accomplished by conduction alone. Consequently the intuitive feeling, expressed in Section 4 (ü)» that -tlie omission of -viscosiiy will still lead to a reasonable pict-ure of the flow field has recei^ved some
sijpport in this case. Puttingc = 0 does not simplify the problem, howe-ver. 7 • Gjanclusj-CTis
The processes v^iich take place in a gas, ini-tially in a imiform state, vdien it is placed in sudden con-tact wi-th a solid at a different -temperature have been examined for "two -values of Prandtl nimiber, namely I: and zero. The characteristic time for the establishment of -the resulting flov/ has been found to be of the order of the mean time be-tv-i-een collisions of the gas molecules whence, since the formulation treats the gas as a continuum, the main effort has been concen-trated on finding solutions valid for large times.
In -these circumstances it has been found that the flow field di-vides into -t\"/o regions in a rougli sense. Sor-ie distance from the interface tlie dis-burbances propagate out into the gas as a v/a-ve motion tra>.-ve lling at the ambient isentropic sound speed (in this linearised treatment), upon
v/hich are siiperimposed the dispersive and dissipative effects of -viscosi-ty and heat conduction. When the gas is intially hotter than the solid the wave front is of an expansive character which tends to flat-ten as time proceeds. If the gas should be colder than the solid initially, the v/a-ve front Tdll represent a compression. This front will still flat-ten and decay vdth increasing time in the linear theory presen-ted here, but in practice non-linear convecti-ve effects v/ill oppose these processes and a shock wa^ve may be ejqpected to appear.
In -the regions near to the vra.ll the heat diffusion processes dominate, but, as "the -tempera-ture of the gas changes, corresponding changes of pressure and velocity v/ill occur and these gi-ve rise to a wave motion v/hich is
superiniposed on the main process of diffusion.
Baebuild up of the flov/ field can be explained as folloirs. Assuming the gas to be hotter than the solid, at the initial instant, the layer of gas molecules immedia-tely adjacent to the wall lose some of their energy and momentxmi to the solid, but as yet there has been no time for any
appreciable mass motion of the gas to occur and the densi-ty remains at its ambient value. The pressure pulse so produced -then begins to propaga-te out into the gas, dropping the gas temperatxoro below that which could be at-tained as a result of heat diffusion alone, and accelerating the gas towards
