!:~ 7'&/'f~1M-\~~~'

('

.~."

.,~. - ..

.~.

,,'~ ....... _.~ '",' ..-, ..... , • _ ..... ". _c_"i.. ........ "-........ _w .~. ,. I ,., .,_~.....__ .....,,'~. " •.. _'"• • ,.• ,_........ " _.......... ~. " ".&' -, ,...~ . "., " ,',-, .. -" .' ,iliI\llll II1II !\II\ III 111\\ U!!lllll\U\!lllillill

TRP0070S

,;;g z: ,OMU, , .• _,A.U]:;:;:

:.:'

.--t

..

LIST OFSYMB9LS. •

1.

2.

3. ANALYSIS. •

3.13.23.3

4.

5.

6 •

•

equilibriu;n.

variation of

y

c.

Fi'.:\.lrl:.! 1. Shock Induced Flow ..On

enthalpy

layer.

In reference [1J,

appropriate equatfQns.

MWRfirst approximation wClS accuratet6'wH:hin a1:>outf.lper-

cent of the

Crocco {31, while a· second. approximation agreed

than one percent. It is on .this basis that the

cated problem to be treated h¢re(Le., including the. /': :-..:,:'.. , ' .:'-'.'. "",> . :'.:,,:,,-,':',' "";-" ,,~,

edge effects) will incorporatetheMwR technique 1:n asecond

approximation together with a fini~e diffe;ence method for

analyzing the. comp.lete unsteady flow.

...----_.._--------.....,.-~---- -----,,--

In

leading

leading edge

ahpu-+dX

Energy

ah::''3t t

The thermodynamic l:i.nd transport.

ass~~ed in the form:

The boundary . conditions for.

, .,' >'.:: <;<;~' ~.".,'.

.. .. ···:/0: .;____,__.-_•.,-'.('.':,: I.....

Pr

h constat'!":w

lJ =.

p p(h)

S'lOC;~ tube flo~..s. The initial and leading:-edgeconditions

capacity of the splitter plate and the short

This last ass'.m1ption can be justified by the

the transverse velo8ityperperidic~lar

tively. It will also be assumed tha.t

remains constant durinq thefloW~

where u is the

-

are:

u(x,y,O) 0

v(x,y,O) 0

h{x,y,Oi h (2.6n)w

u(O,y>O,t) U (2.60)e

v(O ,y>O ,t) 0 (2.6p)

h(O,y>O,t)

I.t is convenient to· normalize thedepenaellt vaziables. .

in equations (2.1) to (2.6) .with respect t<?their .values

the freestream, and to

variables as follows:

is'''' ::

'f -'

(

lJ t

;'\~::,j:,;/'( ,-''-':~''.:'-:: .::~,:

. ~ :-' ;~{,/,::;:,;: -:~,;'~:':::~'::,:::~~ ~./ :'.

; """-'<.",~>-, ,,-':. "

Ii, 77

u*(f,;,n':"tx>fT) ..

u* U;:s,n>O,T)=1

h*(i;:><;s,n~(). '" hw/heh*(;,O,t) =hw/heh*(i;:,n-·"',T)':" 1

h* (1;.5,1'»0,1') = 1

dP* ap*u*.a-:r+ay--+

dU*p* dT

where Lis anarbitraryrefererice lengtn. Under

transformation,. the governing

Conti.nuity

Momentum

Energy

£

Initial and leading-edge

U*(~,!1,O) 0

V*(~,TJ,Oj 0

h*(f;,TJ,()

Equations

tial equations

T.

ll] for

subject

initial

through

3.

In

conditions (2.8) to (2.10)

in the vicinity of

weighted residuals or MWR co, reduc,e ,the n~erof'ihaependent"--------",~-~-,---~.,,-'--"~-,-,--~,---'---

variables from three to two. The resulting caic:ulat,ionsfor

an MWR second approximation were COlllpar.ed ,W'ithMir~+s,sol~-...:;;;i? ,,';,'

tions [ 4 J for a perfect gas and showed 'agre~n1erit'~t thin one '

this close agreement, the MWR,

[1J will be employed to extend the', analysis to, the

edge problem.

The full details oftne MWR analysis are given in

[lJ and will not be l:epeat~d here. Briefly,hoWever>tlie'". -.- , , ., -

following steps are taken: Equations (2.8) ,and{2.9) ar,e. . ' - ,- , .' .

combined by suitably employing a weight1ngfunction and the .' ', . . " . .

::esulting equation is integrated acrosstheboundarylaye.r.,

The independent variable nis changed t.o uby

function

Sirni.lar steps are taken with the energy equation', (2.10).

The dependent variables which appear in the resulting equa-

tions are then represented by an N-th order apprmdmation.

For a second approximation, N;2, the expressions become

p*8 (~.1a)

(3.2a)

r :lC'" ')0 )1'

:oJ .

. +.1

L(Y 1 ~ 1) ']

ao ho hw/he

a l -3h + 4hl - 1

0

a 2 = 2ho -4hl + 2

r('0','l r-: ~1l(~l~l)'J l~-~J

and hI is determined from

srecified solely from the

The parameters 0i6i and hi for i

of T and ~ and are dete~mined by

equations forpo 8o

"!here

become:

for i

2+-

3

0,1 and

independent variable which has a finite absolute ra~ge,

zero to infinity. Fbrthis

ulate the governing equations by finite

tional standpoint,

the flat plate toa value wtiic~i~ gener~lis dl~ti~rJmiriedbl"'.,"·'the arrival of the

.,-"".;::" -

where theconsbnt coefficients Ai by .' .

equation ( 3.3). . Howet,er , it is not 'possibleto~ Ob1:airisuCh;',:'-,:-' '-"":-:--': .; .. .'.

'either~ perf~c:t or realgas.W:henthe i~adiri~·. <,' .'. '>~",>,' --": :': ' '... , '. '.- _. ,-.- '.,

edge condi.,tion( 3.5al is employed·; ··;£h\,1s;.· .<i'. :ntimerical·appro~cb.<

will be taken and . soluti.ons will' be ~O~9ht byc:orisidering'all.(..

appropriate finite difference. technique. '. , '>;' ..... :.'.;..... ,:-

" ,-' ..~.- '::~,., ~ '.

3.1 Finite~ifferen~e Formulation" "'>':~~'.The· range .of the independentv~dable~(ma)rbe taken to

However, the nondinumsionalizedtime variable ~~kes'from zero when thesho~k ~ave a~rivesatt~e' leading' edge

Equations (3.3 )an9 (3.4) arencnlinear partial differen­

tial equations in the two independent variables T andf,. It

was shown in reference [11 that for a perfect gas with <Po

and ~lConstartt, it is possible to obtain an analytic solution

toequationthevi9·inity this,

(3.7b)

beyo~~ ~hich no tlow ~X1S

f;n+l = Sn + lIf;n

;,;.-oi nt

Nmax '" IlL'; + 1 (3 • 8)

will be recalled that at 1"'0, the shock wave just arrives at

the leading edge, n"'l, dPd as the flow ?rogresses ii.e., at

in which the first node is located at the leading edge. It

the length L of the flat plate is given by

and

at all the nodal points.

Let the nodal point l"ocated at· then-th position have a.

coordinate (~n'::) given by

(for n"'l,2,3, ••• ) will be deS'ignated the neighborinq no.d~s

of the point at tn' !nequations (3.7), thed~stance

between the n-th ar.d the (n+l)-th nodes is equal to the

local step size ~~n. !n general, the internodal distance

need not be constant, but in ,this study a constant step size

will be assumed. Thus the maximum number of nodes spanning

Then the nodal points given by

to treat the resulting difference equations as a system of

simultaneous ordinary differential equations for which a

solution .nay be found by solving for the dependent vari'l.bles

Thu5 at those nodal points downstream of ns the gas is

"unaware" of the flow; therefore, the nodes for which finite-

(3.;9)

:or .... ~ \,.. , 1

forward difference:

<:'5.··1 < n < 1If;+1

an~

It is evident. from equation (3.9) that

has only right-hand neighbors, and any node

of the moving shock wave has only left-hand

values of Pie i and hi at the node nwill

Piei (1:,n) and ~i (T,n) respectively.

The following equations Clre .the finite

whichapproximate~ eguiltions (.3.3) and (3.4).

of the .dependent variables with

mated by a central difference

difference equations need to be written are given.by

yields the following equations:

Introducing equations (3.10) into equations

+

1 [1 ' ,', .'. ..,-.;:-" --1~5'0 9 .( t,;; }+;'019,' (-r';;',)l,"['·'''''''h' '~.,' .>~~... \J n:l • ", n.J • . n'tl

and

and

wave. Because the

for which the nodal index n iS,greater

tion of equations (3.12) and (3.13} with respect to

performed for ns -2 nodes. It is clear from 13.9) and

that no integration needs be performed

.,.,.':. '.~,

.,,'

'~': ·:'·':"·.~'-f',;~-: ;' ,,' :<'~ '. -<_<":-;:.:~ ";:

_ ..M.....

>~,' ,':', ,::"

of the .noGal indeX.

'''. ",' "--', .. .-, " ,', ," ..:, " ..';.....

It foUowstherefore that at the end of .ininte~ration~ .

It maybe noted that themagnitQde

x '= L.

n=ns increases as the shock advances from its initial t:>0sition

at the leading edge, n=l, to its maximum value given bYN~axin ~uation (3.8) when the. shock wave arrives at th.e.. p~irit ...

n=n •s

,.

.',.-,;'

. ;,-;"".:",.,

.... ',:, :',.

A difficulty Wh~Ch needS~o 1::leres9·lve4 in starting the

reduced momentumaridenergyeqQati9ns~the

given in equation (3.lSa) with~iei(~,OJ=O are Unsatisfactory

for starting numerical' integration. Thus sever.al featQres of

this flow need to be considered in' order to obtain a physically

meaningful starting formulation. In addition to the continuous

axial expansion of thE:! bounda:r:y;.,layer flow on the plate, it may

be noted that for small times the distance between the leading

edge and the foot of the sho<;k.wave is in general small.and

its magnitude. is determined byt.he velocity ·of the moving--_.......... . ' .....

shock wave. This is an important feature of ;:his flew which

solution of the unsteady' problem in tbeentire.region

19

distinguishes it from the similar problem of a flat plate

suddenly accelerated or impUlsively moved in a

fluid initially at rest. In fac:trf"orthe suddenly-ac:celer'at:ed

plate case, all gradients of velocity and .tempetature are zero

before the inception of the flow, . but these gradients; are

finite for any lima11 time greater than zero since the flOW,>;

exists ever~.ere on theplat~ (th~giadi..entscanbeestiIn~ted/}·;·.... ,.' .

from the related shock~vicinityproblemin coOrdinatesi'ixed ',",,';i.".",.,,:

. ,':.

<c'; -~ -,; ....: " ,,' -:: "-.~.-

In the.

..',' ' ..- ,,-

the· dependent. variables p. e., .. 3.. 1-

shock-vicinity or "Rayleigh"can be obtained.in the

the flow will "feel" the leading edge). It

to propose that, after the shock wave moves past the

edge, the flow characteristics of the boundary layer at the

leading edge and statiOnS downstream of it. will develop con'-

. .

accurate evaluation of

. .

of the rlow. In the present leadi1lg-edgeprqhlenl, it

possible to idc:ntify ~"Rayleigh"regiOnwhich at any

ends at the foot of the moving shockwave and extends

at least to a point where ~=, (further upstream of t 1,is point,=

wave~

relative to the~hoCk wave. tIle "~Yleighhpr?bl~}.present shock induced leading-"edgeprobiem, however,

are the dependent vari~les to be Elstimated f6r ~mall

it is Clearthat;tbe estimate must· be made in. a S1'lort;

tinuously with time. At some intermediate p<:iint, the regicm

:.,,~.. , ,

,~:~~:·.s,:,,:,·,:, -::.:'~' -.. "/.

' ..:":,,.";;'

of the moving shock.

.. ",: :---.::, '<:" .. :- ''''''~'-, <-,.';;,'::".

blend into the region of the flow influenced by the locat:ion·

con'dit::i.on. ThiSi$ ..~. direct result, ofcourse,of.ih,~:.bOun"';.: .""",:,'..>'

~;,: ~ ,

dary-layer eqUationSbeiri9pai::ab6licrathe~thaI1eiii.~t.:ic>- ..•..

partial differential equations:." .. :' , , ',',

(3.16)(~ST_ ~]_ 1 e

'", ,', ','-

In this study, the valu~s of p. 6; were estimatedinthe1.1

-Rayleigh" region by means·of·theanalytic perfect9~s solu-:-

tion derived in reference [1], thati~equation (3.6),· t0gethe.r:

with equation (3.la) and the solution given by

modified by Lam and Crocco· [3}, in the form:

Specifically, the constants Ao and Al of equation (3.G) were

determined from equation (3 .Ia) evaluated at u*=O and '.1*=1/2,

respectively, with the left hand side of (3.1a; assullled to be

given by the first approximation

p*8

2.1

p.. e ..11.

The starting conditions

tions required

and·w

(3.18) •

for computational reasons, but properly

edge and nRayleigh" region variation for the initial

only provide the nonzero values vi

evaluated using equation

for A., substituted into1 .

starting values for

T<~<~ i for- - s .

of

but alsobecause.the time and longitudinal space

are constrained such that

or in nondimensional valiables

T = 1';s(w-l)/w

The time increment is thus obtained as

{3.2.l)

6T. := "'~(w-l}/w (3.22)

Hence r by choosing lit; r In is determined frorr, (3.22). The

0.01

23

choice for ~~ is made such that the inequality (3.20) is

satisfied at each node point for all time. A value of ~~

was round to be the most satisfactory.

In the physical domain, with ~~ = 0.01, the corre.spondinq

time step size is equivalent to the time it takes the shock

wavef~?nt to traverse one ~ercent of the total length L;

.While this constraintll\ay. be •• too severe in. the case of weak

shocks, it has distinct advantagl:!sfor high iritl:!nsity sho.cks;

for increasing shock intensities, the time step size is

decreasing, thus .. assuring a reasonable time resolution of the

rate of growth of the dependent variables at each node. Per"­

haps the best criterion for stability of the solution is given

by equation (3.20); in this study it was used as· the basis for

discarding those step-si.zes which yielded inconsistent results.

The computation for the time dependence of the variables

poeo ' Ple l and h l at each node was obtained by inteqrating

(3.12) and (3.13) repeatedly for all nodal points n in the

range 2~n~ns-l. The node ns corresponds to the current location

of the shock wave, and at the end of each integration nsis

increased by one. The node ns-l always lies in the "Rayleigh"

region, therefore its starting values can be estimated as

described above in Section 3.2. The resulting calculations,

then, are a combined MWR second approximation and finite

difference solution. Results were obtained for the appro­

priate boundary-layer parameters and are presented in the

~ext section£or both a perfect and a real gas.

4 ~ CALCULATED RESULTS

temperature of 530 o R. For the perfect gas calculations,

(LIa)c: o

poe o

Skin-friction coefficient:

The standard boundary-layer parameters may be computed

from the following definitions:

<p=l and Pr = 0.72 were assumed.

reference [1). All of the resulE~Eeported here were made

assuming an initial pressure of 0.001 atm and an initial

Ahtye and Peng [5) as curve fit by Marvin and Deiwert (6):--~---,-----=------------=:-::,,:-,,_,,,,,,:,,,~.again the details including tabular values, are given in

nitrogen using the transport and thermodynamic properties of

The solutions for poeo ' PIEl i , and h1 0btained from the

finite difference equations (3.12) and (3.13) t>Y the method

described in the last.section are. substituted into equations

(3.la) and (3.lb) to yield theappro~imate solution for p*e

and h* as a function of Tandf;.. In 0i::ltainingthese calcu­

lations, the thermodynamic: and tr:ansport variables p*, $,

and <p!Pr are related tOll'll by polynomial curve Htting with"

the coefficients evaluated by a collocation procedure: the

details are given in referen.ca [1]. The. th~~.dynamicequi­

librium real gas calculations reported here were made for

in figure 2 fo.r the perfect gas case, and in figure 3 for the

25

(4.lb)

*UeL/Ve and Ue is arbitrarily .taken

1 (<lh*)eo au* u*=o

. *Ue

= ! p*eu*(l-u*)du*o

6*L

Nu{l-ho )=ReL

Momentum thickness:

o

Boundary-layer thickness: *Ue

i~= 1" e du*LL

Velocity profile:

f IReL

= [U*s du*

o

Displacement thickness:

Nusselt number:

Energy dissipation thickness:*U

6***. =-:-- (e 2L >'ReL - J p*eu*(l-u* )du*o

In these relations, ReL*as Ue =0" 9 95 •

The development of the boundary-layer thickness is shown

5.0

oo

o Shock-vicinity Solution [1]

--- Present Solution

.1 Location of Shock. Wave

4.0

3.0

2.0

1.0

o 0.1

fi.gure 2. Boundary--Layer ThicknessPerfect Gas, Mg ",. .3.15.

,.

F.i.yun,

ol-_.....,.J...__.J.--:--~_~..;..L..,~~L-:..~+~~~~q~_..a-_-oJ

o 0.1

~---------------~..-.....,-.,..--

case in which the transport and thermodynamic properties are

evaluated from ~e real gas properties tabulated for rtLtrogen.

Figures 2 and 3 show the manneiof propagation of the lead.irig

edge effect as the shock wave moves down the-plate. The

region affected by the leading edge asymptotical~y.increases

with the downstream movement of the shock wave. This obser­

vationappliestoall intensities Of the moving shock The

comparison shown in figures 2· and 3 showsagreementwithtbe

shock-vicinity analysis of reference r1] in the region cl"se.

to the shockwave. The deviation betweentbe two soiutions

increases with distance away from the shock. The reason for

the unexpected waviness in the calculated curves downstream

of the maximum value is not known at present, however it is

probably a result of the numerical smoothing procedure at

the T=~ point (this will be investigated in future studies).

Figure 4 shows the boundary-layer t.hickness parameter at

that instant when the moving shockwave is located at 0.99 •

for different values of the shock wave Mach nUI!ber, Ns

'

Unfortunately, the boundary-layer thickness parameter give~

by equation (4.lc) tends to conceal the physical variation

cf the boundary-layer thickness c with shock wave Mach numbe::

MS

because of the manner in which the Reynolds number

ReI, = UeL/ve enters the calculation. '1'0 illustrate this

point, tigure 4 is replotted in fiqure 5 by assuming L is

one foot and calculating the Reynolds numb"r .fc)l the apprc­

pr iate real gas properties. Figure 5 sncwsthat the bOU.DC.ccl-;

layer is in fact becorni:1g thinner with incrci'lsi !HT sr,":'<.Tk

intensi ti.es ~

,!fir _",;_--','

\ic

.,6 0 b

:,

= 12.,?

5 .0 l'"

:;:.~

;,4 .0 n

:11-

~~ 1~;

'<Jl...:l Jt

".')

1.0

o

Figure 4.

,\

...__.._----------------_._-------_.-.....-._._...._-----_....._---> ..... -;>,'.:":~ilti1fIIIIM:. •• iii _ =-= - -

r' r'

)

"

.)r0,

j

t,),t

o.

J,)It/,t

_..---->...-..-.----_.-....._~- ..-.......:..,,

o .•__.-" L _ 1_.. _ ..J,_....•__., -.1. _ ...•.._1. __.~t ,..;.,1 ·•• __ _..,..:-..;-I......;....,..."..

o 0.1 ~.2 0.3 0.4 0.5 0.6 0.7x(ft)

Figure 5. Boundary-Layer '1'hickness as a Function of $hockWav~,Int:en~;itY(Assuming L = 1 ·ft.); Real Gas.

2.0

1.5

0.5

<5

(H)-3 ft )

1.0

I t may be •~:e:.:d::...::.:=-=-:==::....:::....:==---.:~==-==-=--=::::...==solution, accounting for the leading ed9~,is

"', ','.: ,", '

different than the shock-vicinity solutfono,freferer-ce tIl.:,',",',;:" : ,','. "",

except qUitecloseto'the,sbockw.ave~

curves -,are def~ed-as ," thei()~~sof ,isoi~ts·-in-spaee a11~_•.ti~, _-­

at which the -bo~ndaty-layerth~cltness has, reached~S.perc~_nt

of its' steady state value. - Simila~resdlt~ have been-':"'~~'----

obtained experimentally by F~1~~~{71and~aviesand

Bernstein r81~ased on wa~l heatt~~~fermeasurements•. , ,.' ,.' , , , ~

present results are compared with 'the experimerital values in

figure 7 where it is seen that the agre_t is v~ry satis­

factory over most of. thE! range

Figur.e 8 shows a comparison of skin friction tesults. " ' " , ' , , ". '-- ,. "'-"

-""" "

for Ks = 1.6. Again theagreenient with the sh6ck~\Ticinity-""~.~

Odel is good near the shockwave an4 a progressively increasing

departure is observed as the leading edge is approached. It

is also seen that, .l.S expected,th~e~~no.essential_

'-between a perfect and a real gas calculation. Figure 9 shows~,~"•••__ ._,_:, __•• "O ."__._. ._

the skin fric1:io.1 distrib'..ltion for two post tionsof the

moving shock wa-,;e for a perfect gas a.n.d Ms '" 2.2. Ir:, fig>.:"::,,,

.;;.~.-: ,:~:'''~';: .',". '..:'.,','.'....-

",;.-

'...:';'.'.

"

Figure 6. LocusofP.oints· in Time and Space at Whichthe ~oundary~LayerThickness Reaches 95Percent of its Steady State Value; . Rear Gas.

··-!fij~""~~~IJIIia1I!11l11Dlnllll].lrll J --"'".---------'---------------,..----"'"

.. Felderman (7)

6543

··--·--.,.-·-r--·T--,----T..-T------'-··r--·-.----r··-·-,----r·-----.,.-~

EXPERIMENTS:

• Davies & Bernstein [8]

Figu.r;e 7.

8

7

6 ...

5Uts 1

K 1 4

3

2

1

0

1 2

1.00.8

Comparison of Skin-Friction Coefficientwith Shock-Vicinity Solu1;.ion(lJ forM = 1.6. .s

o

Figure 8.

0.80.6x/L

o

Figure 9. Skin-Friction variation Along thePlate for 'l'WoLocations of theShock I Perfect Gas , M =::.2.

s

o

1

location. Since the hump shown in these figures, which obviously_•.,.~: --"-'-'--'-'-__~,_.-c"':;_~'_,;,_~ ~_:~.::•. _

illcreases with increasing .. values of. M,was not encountered in .·_·_-"~-,··--;~·_---.,-·------.-,---:--·-s·..·_··, .. ·- .,---~ - -- '" '-_~~: .•.. _

the shock-vicinity calculations of reference U J for the

10 a comparison of the skin friction results

and the shock-vicinity analyses is shown for a

w.Lth Ms = 3.15. Figure 11 shows

assuming a rea.l gas for both the complete·and

vicinity flows; it may be observ'ed that the stlo<:k··v;iC::inity

results deviate s1igh.tlymoreth.Cin wasnoteCl..

The comparison. Shown iilffgure.I2 ls·for the

friction for both a perfect and real gas with Ms = 5.•

results show that the perfect gas model now yields a

value of skin friction at all points on the piate.

The evaluation of the heat transfer at the wall is based

upon the Nusselt numb~r..a.s.~.~~=~..~Il~c,ttlation. (4.19). Figure-----_._.--..---.-..- ...._.. . . . .

13 shows the·Nusseltnumber variation for both a perfect and

real gas with Ms = 1.6 for the complete and the shock-vicinity

solutions. It is seen that the shocko:-vicinity. solution departs

from the complete solution more than was observed previously

in connection with skin friction va.riation. A comparison of

the perfect and real gas solutions for shock wave Mach numbers

of 2.2,5, and 12 are shown in figures 14,15, and 16, res­

pectively. It is clear from these figures that the real gas

departure from the perfect gas calculations becomes very large

with increasing shock wave intensity. However, these figures

also show that the present solution apPears to be encountering.

numerical stability problems downs tre arn of the shockwave

-_ ...--------_.-

4.0

~~'H 3 •. 0u

••s ",IP-. ---r

0.2 0.4E;

Figure 10. Comparison of .Skin,..FrictionCoefticientwithShoc)(.,..Vicinitysolution [I,):Perfect Gas,M ... =$ .15.

s

•• .5

i

1..00.80.40.6.f; = x/L

Comparison of Skin-Friction Coefficientwith Shock-Vicinity Solution [ll:Real Gas, Ms = 3.15.

0.2

Figure 11.

a Shock-Vicinitysolution (1)

0"-__- ......-..-......+-......--_..1-_.......+_.....o

2

1

6

7

__=.i ~;

Fig"'l:e 12. comparison of Skin-friction Coefficientfor Perfect and Re.,al Gas Solutions,Ms "?'

Figure 13. CompariSon ofDistribution,

o 0.6

x/L

0L.._...,........__~a...;.__...r..__--I .J

":;"~ ,~.?, ~c1- ~ ,A'~-, .- ~~-~ .~~. " N

41

Perfect Gas

- ---- Real Gas

tI,i

I1~.

7.0

Figure 14. Comparison of Nusselt NumberDistribution, ~s = 2.2.

5.0

6.0

I4.0 f

~H fI

"- ,0 3.0 I.r::

II.-t

I~ Iz 2.0 ,

II

I1.0 ,,/

""""" ."...---

0

0 0.2 0.4 0.6 0.8 1.0

S x/L

Figure ·15. Co~parison of NusseltNumberDistribution, Ms=s.

0.4

~=

0.2

--- Real Gas .

o......---'-----........--_.....ll-o.. ............~""""""o

16.0

l:i:.O

:sz

.~~.....

o.c,"""..........

I

.:-

r

I Perfect Gas

II

iL , ;.

43

80.0 -- -- Real GaS

60.0

1€40.0

"-0.c;I~.....~z

20.0

o

III

. IJJJ,,II,,, ', I

\ ,. ....\ '/ \ I

\ / './\_"

o 0.2 0.4 0.6 0.8 1.0; x/L

Figure 1.6. Comparison of N·.l~selt. N'..lmberDistribution, M ~ 12.

s

rJ

it

corresponding shock intensities, the cause probably lies

with the numerical smoothing procedure discussed earlier.

Again, this point will have to be investigated in more

detail in future work.

£

flow

wave moves farther andfarth~r

well with experimental results

measurements.

Results were also ShOWh for the

and the Nusselt n~~er for a variety

.~1!

"'

-.,(-.

r