AIAA 92-0806 VIBRATIONAL RELAXATION OF ANHARMONIC OSCILLATORS IN EXPANDING FLOWS

S.M. Ruffin and C. Park NASA Ames Research Center Moffett Field, CA

30th Aerospace Sciences Meeting & Exhibit

January 6-9/1992 / Reno, NV For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 370 L'Enfant Promenade, S.W., Washington, D.C. 20024

VIBRATIONAL RELAXATION OF ANHARMONIC OSCILLATORS IN EXPANDING FLOWS

W Stephen M. Rulfin' ChuI Park"

Acio~licriiiodynaiiiics Bt a~icli NASA Amcs Rcscarch Ccntcr

Mo Ifcu Ficld, Cali rortii a

ABSTRACT

Although the Landau-Tcllcr vibrational model ac- curately predicts vibrational cxcitation process in post- shock and compressing flows, i t undcr-predicts tlic rate of de-cxcitation in cooling and cxpanding flows. In {lie prcscnt paper, detailed calculations of tlic vibrational relaxation process oCN, and CO i n cooling flows arc con- ducted. A coupled set or vibrational transition rate cqua- Lions and quasi otic-dimensional fluid dynamic cpa t ions is solved. Multiple q u a n t u m level transitioii riilcs arc computed using SSH Tlicory. The SSH transitioii rxc rc- sults arc compared with availablc cxpcrimcntal d a l a and otlicr theoretical modcls. Vihralion-vibration cuchongc collisions arc responsible for soiiie vibrationd rclasation acceleration in situations of high vibrational ccmpcraturc and low translational temperature. The present results support the relaxation mechanisms proposed by Bray and by Trcanor Rich and Rchm. Qualitittivc agrccmcnt with cxpcrimcntal results is achieved lor tlic overall vibra- tional relaxation rate, however the accuracy o f the SSH results for vibration-vibration cschangc tr;lusitiotms must bc studied lurthcr and ndditional cxpcrimcntal invcstiga- lions arc ticcdcd for quaiititalive 'I ' g rccnicm

NOMENCLATURE

A nozzle cross-sectional area cvib

E,. f(5, h Planck's constant H static enthalpy Ho statnation cnthalpy k Boltzmann's consmit

vibrational energy pcr unit mass relative to ground statc energy cncrgy in lcvcl v pet- molecule Kcck and Carrier xiiabaticity fittiction

* Research Scientist. AlAA Member

AlAA Associate Fcllow

Copyright 0 1992 by !lie American Instituteof Aeronautics mtl Astronautics, Inc. No copyright is nssericd in the Unilcd S ~ c s under Tiile 17. U.S. Codc. Thc US. Govemmcnt has il royally- free license io exercise all rights under the copyright ciaimcd herein for Governmental purposes. All other rights arc rcsencd by lhc copyright owner.

Head, Experimental Acrothcrtiiodytianiics Section,

transition rate cocllicient Cram v to v' molecular mass reduced mass o l the collision pair total number density number density in lcvcl v staiic pressure stabnation prcssure

velocity dcpcndcnt probability of tritiisilioti

lrom v to v ' and h n i v2 to v i

tiicrnially averagcd piohability of traiisitioti

from v to v' and from v2 to v i vector or dependent variables dependent variable in the j- th cqualion intcrnmolccular distance ordinary gas constant universal gas constant steric Pactor intcratomic disranccs time coordinate rrnnslurional tenipcraturc stagnation tcmpcrmtrc vibrational tempcraturc hascd on popitla~ioii at levels 0 and 1 streamwise flow velocity pre-collision relative molccular velocity minimum allowable pre-collision relitlive molecular velocity post-collision relative molcciilor vclocily molecular interaction potcn~ial constant i n cxponctitially rcpulsivc iiitcritc- tion potential highest bound vibrational quantum lcvcl streamwise spatial coordinate

interaction range parnmctcr in cxpo~iciiti~il ly rcpulsivc interaction potential consrants in exponentially repulsive iiitcriic- tion potential cncrgy transrcrrcd from tratislatioti 10 v i - bration per collision vibrational waverunction or IcvcI v density

1

% 7 vibrational rclaxation tinic 0 collision frequency

cffcctive hard sphcrc cross section

r churactcristic vibrational tcmpcratutc =(El-E")/k vibrational transition niatrix dc inc i i i i o r v 'V",",

IO v' iriinsilioti

Subscripts and superscripts E JnlaX LT rclcrs to Landau-Tcller model MW

SSH

V VT refers to vibrntion-trailslation collisions vv refers to vibl-ation-vibration collisions

rclcrs to thcl-mal cquilibriuni conditioti niaxinium number o l dcpcndcnl vnriahlcs

rcfcrs to Millikan and White cxpcrimciilal

refers to Schwartz. Slawsky and I-lcizicld Theory rcfcrs to vibrational quantum lcvcl v

correlations

1. INTRODUCTION

In 1936 [he Landau-Tcllcr ralc cqi ta l imi was derived to describe vibrational cncrgy tr:insfcr in liigh temperature niolccular gases. With vibration;tl rcI;~x- ation times infcrred from posl-shock flows, th is cqitition has been sliown to give good predictions for rc1iix:iIioii in compressing f lows and is widely used. However l l lc Landau-Teller equation fails to predict tlic fiislcr vihra- tional relaxation observed i n cxpcinding or no%~.Ic llows. Thcsc flows arc characterized by a rupidly dropping translational tenipcraturc cind ii high dcgrcc of vibrat ional csciulion. Rclelcd flow l i c lds o f currcnl interest i i ic ludc thrust nozzles and tlic wnkcs behind high spccd vc l i ic lcs such as aero-assisted space transfer vehicles and hyper- sonic cruisc vehicles.

A number of expanding flow cxperimciiis I u v c becn conducted which attempt to quantify tlic iiicrcascd vibrational relaxation rate in nozzles compared io 1pos1- shock flows. A fairly coinpletc tablc of thcsc cxpcri1iic111s for pure gases and gas mixtures of N, and CO is givcn by McLarcn and Applctoiil. Unfortunately thcrc i s wide disagreement in the expcritncntal data on tlic niagnitudc of acceleration in thcsc flows. As discussed by llurlc2, some of the experimental discrepancies arc due io tlic cf- feet of gas impurities, shock diaphragms, and indit-cct v i - brational tcmpcratiirc measurement techniques. I l x h of tlicse expcrimcntal errors may enhance thc ovcr;iII rc- laxation rate. However even the most reliable cxpcri- ments indicate that for pure diatomic gases and sonic gas mixtures, vibrational relaxation is accelerated i n cspnnd- ing flows.

Thcoretical models proposed by Bray3 and by Treanor Rich and Rehm4 indicate that tlic higher rclax- ation rate may be caused by overpopulation in niciny of the vibrational quantum levels. This ovcrpopulanion is

due to tlic effects or rapid vibration-vibration ( V - V ) cx- change collisions rclativc IO the vib~ati~ii-tr ; insIaii~ii i ( V - T) collisions. A V-T 1r:insiiion corrcsponds IO 3 collisioti of rnolcculc 1 and parliclc 2 i t i which pariiclc 2 is cillicr :in : i to t i i o r ii is :I nioIcci~Ic t11:it docs no1 1i:ippcti IO cli;iiigc i t s qtiatituiii SLIIIC as a rcsiill of tIi;ii collisioti. 111 V - V c x - cli;tngc collisions, one tiiolcciilc gains :I (Iit;In~utii of v i - brational cncrgy and ~ l i c OLIICS looses otic q i i i i i i ~ u ~ i i of vi - hr;itional cncrgy. Each o f tlicsc V - V cxchnngc rates oc- cur with high probability bccausc only ii sni;iII ~itiiou~iI of energy is transferred bctwccii vibration and iranslatioii. The probability of V - V collisions in which both molecules gain a quantum o f vibrational ei1crg.y or in which both niolcculcs lose ;i quantum of cnergy IS much lcss than tlic probability of V-T or V-V cxchangc colli- sions. Tlic Bray model l i i is bccn shown to give qualitalive hut not quantitative agrccnicnt with cxpcrinicntal data.5 IHowcvcr this model assumes a ft!nctional form for ihc population distributions and considers o n l y nearest ncighbor iriinsitions. The functional lorni used by Bray was proposed by Trcanoi ci. al . , and neglects rapid upper lcvcl V-T traiisilions. Tlic quaiitiiaiivc results may :ilso be ;il'lccicd b y use o r a simple cxprcssioii for t l ic transition matrix clcmcius rather than solving Schriidinycr's equation.

In tlic prcscnt paper, dctailcd calculations of the vi - brational rclasation process of pure diatomic gases in i i o u k and cooling flows ;ire conductcd. X2 aiid CO arc studied at translalionnl tcmpcraturcs Icss t l ia i i or cqit;iI IO

4000 K. A coupled set of vibrational trmsitioti ralc d c q i ~ ~ i ~ i ~ ~ n s and quasi 1 -D nu id dynamic cqualions i s solvcd. Transition r;iIcs arc conipii~cd i'roni SSH Theory ;uid iiiiiltiplc quatitimi lcvcl V-T iind V-V lrmsilions arc allowcd. This paper sccks 10 cxiiniinc Ihc col l ision pro- ccsscs rcsponsiblc for iiccclcratiiig vibrational rcliixatioii in cxpandiiig flows, prcscnt coiiiputcd population distri- butions, and invcs1ig;itc several aspects O f SSH ciilculii- lions.

ur

11. VIBRATIONAL TRANSITIOK RATES

S S I I Formulalion SSM Tlicorv w w dcvc lo i~cd IO allo\i. colcularion of

transition rates for niolccular yascs. This theory is dc- scribed in dclail by Scliwartz, Slawsky, and I-Jcrzfcld6 (SSH) and by Clarke and McChcsiicy7 and wi l l only IIC oullincd it1 this paper. SSH Theory iissu~i~cs cnd-on, e o - incar collisions of two rotaiionlcss particles (molcculc 1 arid particlc 2, say) wi th cxponcntially rcpulsivc IpoIcil- ti;iIs:

v = vi, c~a('+P,s,+P2.2). (11-1) Tlic collision partner, particle 2, can cithcr bc a niolcculc o r an atom. The distance bctwccii ll ic centers of mass o f tlic IWO particles is given by r and tlic intcr;ilotiiic dis- t;iiiccs o l particles 1 and 2 iirc s i and s2, rcspcctivcly. a. L/ p, and D l arc niolcculnr c o n s i m t s wliicii will be discussed

in the next section. Consider I)articIcs 1 and 2 init ially in For exact rcsonancc cascs, i.c. AE = 0, the probability vibrational quantum statcs v and v2. rcspcctivcly, and with prc-collision rclativc velocity uo. For cilcli collision

,,,ill ~ r i l l l s i l i o l l IO statcs v' and v ' ~ , rcspcciivcly, is

v .v'

pV:. proachcs o i l ic rcsoni~ni probability bcconics

16x2m,kT p"',?2 = v , v - , \'L,\'2

is not wcll dclincd, howcvcr in the limit as AE ap-

pnrticlcs, tllC proba~)ilily t ~ l B t

(11-7)

This cxprcssion is uscd 10 coniputc tlic tratisitioti prob;r-

v.v " V' 2 2 /I,,;;. = 'q,. 21z.'.2

wlicrc 47C2111,,U11

4TC2111,,ul; a h 8,)

0,: = all u,: is the post-collision rclativc vclocity and ni,, is lhc rc- duced mass of the collision pair. The transition iniairix clcmcnts are given by

c.2

q,J j.," . G J w,(s,) c ~ ' x ~ i w,.(sI) (is, (11-3)

whcrc as i n Sliarnia, Huo, and Park#, w, and w,, arc f o u n d Tram nuiiicr ical sol ii t io n s o (. tlic one- d i mens io na l Schrijdingcr equation of vibrational niotioti muming an analytic Murrcll-Sorbic potcntial. Bccausc w e ncglcct rotation, dissociation, and radiation, i l ic transliitional cn- crgy lost during a collision niiist all he trandcrrcd into ~ h c vibrational mode. The cncrgy transrcrrcd from Iranslaiion 10 vibralion is thus

-03

AE=(E, .+E, . ) - ( E $ , + E \ , ) = ~ n 1 + [ ( 1 + ~ ) 2 - I ( L I J ~ ] 2 2

(11-4) wlicrc E, is tlic cncrgy in vibrational st;itc v pcr iiiolcciilc. Inlegating ovcr all pcrniissihlc inilia1 velocities wliilc as- suming a tlirec-din~ension;il M;ixwclIi;in disiribulion, Icads to the vibrational tt-:msitioii probability:

03

"Omin

I n the present study, two difrcrcnt mcthods iirc in- vestigated to determine this transitiotl probability. Both of these methods have previously been uscd by other re- searchers and i n this study :I comparison of the rcsulting rates is given for two schemes. I n tlic rirst nicthod, tlic integral in equation (11-5) is c o m p u u i nuiiict-ically Ibl- those collisions which correspond 10 vibrational cxcita- Lions, i s . AE greatcr than 0. Thc de-cxcitations proba- bilities, i.c tliosc for which AE is less than 0, arc then round by enforcing dctailed balancing hascd on tlic popu- lation distribution at equilibrium

(11-6)

bilitics for resonant cascs. The second nicthod of evaluating tlic tr;insitioii

probabilitics utilizes an analytical cxprcssion dcvclopcd by Kcck and Carricr.9 Tlic Kcck and Carricr rormula- tion is a curvc f i t for the probabilities integrated over a otic-dimensional Maxwcllian distributioti. Thc proba- bilities ror vibrational cxcitations, A€ greater than 0, vi;) Rcrcrcncc 9 arc

whcrc

The adiahaticity runetion f(5) is given b)

f(5) = 8 4 5 7 1 3 cxp(-is2'") ror 5 > 21.622

The dc-excitation probabilities are again lound from dc- tailed balancing as shown in equation (11-6). Use o l the Kcck and Carricr cxpression is muck rastct- than the nit- nicrical inlegration nicthod and a comparison of Llic lwo methods is prcscntcd in subsequent figures.

We can now computc the transition rate cocfficiciii. K,,,,.. , which can bc thought of as i l ic fraclioii of ~iioIccuIcs i n s t a w v tliat trmsition to sratc v' per unit time per u n i t number density. The transition rate cocrficicnt for niolcculc 1 is round by suniminf ovcr dl possible transitions of particlc 2:

whcrc N , is the per unit volunic number or particles 2 In

statc v2 and N is the total number density ( i t . total nuni- bcr or pariiclcs per unit volunic). The collision frequency (numbcr or collisions or particles 1 and 2 per unit time per particlc) is

2

(11- I O )

whcrc cio is Ihc cffcctive hard sphere cross section and s=1/9 is the steric hctor for the diatomic collision part- ncrs. Thcrc arc scvcral sources or error in the computed transition r a w . The chief sourccs of error include the SSH assumptions or colinear collisions, exponentially rc- pulsivc intcraction potential, and uncertainties in molecu- lar constants such as oo. Bccausc of thcsc errors and

3

uncertainties, the transition rates must be adjusted IO

match experimental data. In this study, tlic computed ground state transition rate coefficient f o r V-T transitions is adjusted to match the value infcrrcd fro111 shock ruhc data compiled by Millikan and White”’. In shock tube experiments tlic Landau-Tcllcr cqii;lrioll I S uscd with good succcss to prcdici the I ~ S I shock vibrational rclaxation The cquilibriuni ground sr;lic transition rate can bc infcrrcd f r o m the Mil l ikml atid White vibrational relaxation time by using

( 1 1 - 1 I )

For tlic present calculations, oo is chosen so Lh;it i l i c coi i i - putcd equilibrium ground state transition rate imiiclics tlic Millikan and White value. Thc oo thus obtained is uscd for all of the V-T and V - V transition rates. A similar correction proccdurc has previously been used hy Sharma ct.al.8 and also by Landrum and Candlcrll i n vibrational transition rate calculations. The rclaliot~ship of all other rates to the ground state rate is obtained from the SSH formulation using the niolccular coi isI : lnIs described in the next section.

K$= [N z M W ( 1 - c-’vrr ) ] - I

Molecular Consrants Computing the transitioii rates using SSI-I lhcory

rcquires specification of ihc tiiolccitlilr constanis w. PI, a i d bz, The transition rates at-c strongly dcpcndciil on llic value of the interaction range parameter, u., and i irc imiicli

weaker functions of p i and pz. a primarily dcfiiics llic slope of 111c intcraction potcntial as a function or disliincc bctwccn parriclcs I and 2. PI and jj2 dcfinc llic cI1:in:c i n IpoIcntial as Ihnctions of tlic interatomic dis ta i icc \ oc tlic two particles. Approximate viilucs f o r B1 and 13- c a n be found by considering purcly end-or col l is ions i n which only the nearest two atoms of particles 1 and 2 coiririhuw to tlic interaction potcntial. For liomonuclcar molcculcs such as N,, PI = Ij2 = -0 This value is used in ;ill of l l ic prcscnt calculations.

The value of a has been a source of dcbarc and uncertainty for years. As pointed o u t by Bi1liti:s and Fislicrl2, molecular bcam data givcs a in tlic r m g c 3 - 4 A- I Cor N,. Radzig and Sniiriiov13 quote an cxpcrimcnLa1 value oC a=3.16 A-’ for N, and a=3.47 A - ’ lo r CO. However in perrorniiiig calculations of N, transilioii rates using a semi-classical modcl, Billings and Fishcr” rind lhat a=4.0 A- ’ givcs better agreement with V-T lr;iiisilion rate d a h than a=3. I6 A- I . In the original SSH paper, Schwartz et. a1.6 determined a by fitting the exponentially repulsive potciitial to the Leonard-Jones potential. This gives a = 4.72 A-l for N, and a = 4.87 A-’

for CO. Sharma et. al.8 used a value of a = 1.0 A ~ ’ lor N, and predicted a severe bottlcneck in the transition rarcs. Through discussion with the authors of rhat rclcrcncc is was revealed that a = 1.0 A-1 was uscd hccausc ai sonic

given temperature and given cffccrivc cross section, q1, the computed ground state transition rate required litilc adjusrniciit to niatcli tlic ground state transition rate iri- fcrrcd from tlic cxpcrimcntal data of Mill ikan m i Wliiic. Finally, Landrum ;ind Candler’ 1 ohtaincd :I very Ihigli v:iluc of a but have since rccognizcdP i1i;ti this a WP

d

ohr:iincd by incorrectly matching i l i c inlcriiiotiiic Murrcll-Sorbic poicniial to tlic iiiicrtiiolcwlar poIcnii:il. The molecular bcam results and the values obiaincd b y mnicliing the Lcnnard-Jones potential provide tlic mas! plausiblc range or a for N, and CO.

Since otic of our goals is to accurately predict transiiion rates, the best value of a to use in SSI-I calcula- lions is one which allows computed SSH rates to match cxpcrinicntally observed properties. Ideally, wc would likc Ilic SSH rates to match the true 1) ground state magnitude at a spccilic temperature, 2) ground state tcnipcraturc dependence, and 3) the scaling of tlic ollicr lcvcls relative to the ground svatc rate.

There is a vast difference in the amount of cxpcri- niciiiiil data available for V-T and V - V raics. Tlic Millikan and White data providcs a wealth of informatior 0 1 1 the ground state magnitude and temperature dcpcn- dciicc for Llic V-T rates. Experimental data on tlic scaling of oilier V-T lcvcls is sparse hut calculated rcsitlts o f this scaling based on a semi-classical niodcllz providcs addi- t io i i t i l data with which to compare. Thus, in ilic prcscnr sriidy. tlic Zround sLarc magnitude and tcnipcraturc de- pendence of thc V-T rates will be matched to cxpcrimcn- tal data. The scaling of all othcr V-T rates to the ground staic rare will be compared to a semi-classical modcl. This marching proccdurc is described further i n tlic fol- lowing paragraphs.

111 iliis srudy, we imatcli tlic magnitudc of tlic groiind state V-T rate at a rcfercncc temperature b y choosing t l ic appropriate valuc of oo as described in the ~prcvtous section. The ground state temperature dcpcti- dcncc and the scaling of the upper levels arc strongly ill-

rmgc potential, i t is possible that the value of a chosen to liiatcli V-V exchange ~ i t c data may be sonicwhat diffcrcni t1i;in i l ic a which matches V-T data. Thus we will dcriiic iiii a,.,. to be uscd i n computing V-T rates and t i t i lor v-v ratcs.

As shown by Landrum and Candlcrll, a has signif- icant effect on the scaling of the othcr lcvcls rclaiivc io i l ic ground state rate. However, a also has a strong cfrcci on temperature dcpcndcnce of the ground slate V-T prohabilitics. Wc can show from equation (11-8) iliiil i l ic dominant teniperaturc dependence of the ground sI;iIc rates is in the exponcntial term:

flucnccd by thc value of U . Because SS?d uses only ‘I ’ s .I 101’1

. . . The constants h , k. m, and 8,. iirc known wii l i tnittcli grc:ircr certainty than a so we can also write

p:;,$cxp(. T-IC a-213 ) (11-12)

3

Since the Millikan and White correlation gives the ground state rate as a function or icmpcraturc, wc can dctcrminc aVI. by finding the value which gives the same tcmpcrn-

According to Landau-Tcllcr Tlicory, ilic V-T triiiisiiioii rates arc indcpcndctit of the popiilation disiri- butions bccausc PI," = Pi:o = P I , , , = ,.. Thus tlic ground

state probability is found from

" turc dcpcndcncc as thc cxpcrinicntal dara.

0.0 I I 2 . 2

) ] - I (11-13) - 0 0.0 N pi,qMw-in? = [N TMLV ( I - C-'\,"

with l l ic aid or Millikan and White data. The SSH prob- abilities arc lound by two methods: numerically intcgrnt- ing over a 3-D velocity distribution via equation (11.5) or by using the Keck and Carrici- expression, equation (11- S). Figure 1 shows the V-T transition probabilitics for N, at p = 1 atm. inferred from Millikan and While data and from the computed ground state probabilities. For this plot, o0 is chosen so thal the SSH rates agrcc w i t h Millikan & White at T=S000K. Note th:u tlicrc is little difrcrencc bctwccn ihc results when numcrically inlegrat- ing llic 3-D velocity distribution and wlicii using l l ic Kcck and Carrier formulation. As expected, each of l l icse nicthods gives rates which have tlic temperature dcpcn- dcnce shown in equation (11-12) over most of ilic rem- pcraturc range shown. The Radzig and Sniirnov value of aV1.=3.16 A ~ ' and the Scliwartz et. a1 value of aV,.=4.72 A - ' do not quite match the Millikan & White slope. For N,, av,.=3.90 A- ' was found to allow tlic SSH results to agrcc with the cxperinicntally obscrved tciiipcr;iturc dc- pcndcncc and h i s v:iIuc is used for tlic prcscnt N2 calcula- Lions.

Figure 2 shows similar ground st;itc rates for tlic CO iniolccuIc at p = 1 atiii. Tlic SSH results using aV-,.=4.S7 A- ' iigrce lairly well with tlic Millikan and White tempcraturc dcpcndcncc. However, i t is found that aVr=4.60 A-I gives the best agreement with data and thus is used for the present calculations involving CO. Once again the Kcck and Carrier formulation and !lie nunicri- cal integration method give nearly ilic same tciiipcr~iturc dcpcndcncc.

Now, we would also like to perform a similar cor- rection procedure for V-V exchange rates. However, tliere is practically no experimental data on tlic V-V ex- change rates of N, and CO at temperatures or intercst in this paper. Thus, i t is dilficult to independently verily thc accuracy of the ground state V-V exchange rates and tlic scaling of all other V-V exchange rates relative to the ground state rate. Virtually all or tlic V-V excliange transition ratc data for N, and CO is measured in laser systems at temperatures below 1000 K or in C0,-N, gas mixtures at T below 2600 K . A comparison o f SSH rc- sulrs to measured V-V exchange rates i n C0,-N, mixiurcs is given by Taylor and Bittcrnianlj and is reprinted i n

,J

J

4 .

Figure 3. The near rcsonant V-V exchange ratc is sliowti as a runction of Icmperaiurc. Tlic SSH raics f :I 1'1 to accti- rarely match [he magnitude and icmpcrnturc dcpcndcncc o r tlic mcasurcd V-V rates cspccinlly ai very low t e n - pc~iiturcs. The error in ~lic SSIH r;itcs is wi~ltiii ;I r:ictor o f 50. Sliiirnia and Brau16 iiidiciiic tliat f o r iiciir ~CSOII;IIII V-

iiiolcciilar intcruction rorccs wli icl i ;irc 1101 iticludcd iii SSH Thcory bccomc important. SStl Tlicory instead uscs an cxponcntially repulsive potential which only takes inlo xxoiniI sliort range crfccts. Bccausc of i~ncertaintic~ in thc V-V exchange rates, the effects of tlicsc transitions will be studied by using SSH rates and by pcrlorining a parametric study involving various V-V exchange rates and various valucs of avy. In the parametric study, all V-V exchange riltcs arc multiplied by various consiant hctors to study the cfrcct or changes in thc magnitude of these rates. The effect of the scaling of all V-V cxchangc rates relative to tlie ground slate is investigated by using difrcrcnt valucs of avv.

An approximate value of avV is dctcrmincd by comparing SSH computed V-V exchange rates to pub- lished results from the Billings and Fisher scmi-classic;~I model. The thrcc-dimensional Billings and Fisher transi- tion ratc model uses an interaction potential which in- cludes both short range and long range forces. Thus, this model is more realistic than SSH Theory. Figure 4 shows tlic V-V cxchangc probabilities for N, rclativc to [lie ground state rate at T=2000K. The V-V cxcliangc prob- ability initially increases and thcn dccrcascs with quc\iituni number. The incrcasc is due to tlic iiicrciisc in nialris clcmcn~s with quantum number. Thc r a t e then dccrcxc for the upper lcvcls bccausc a i i l i x i i i on ic rcso~ia~icc dcl'cci incrc;iscs [or tlicsc rates. For cxamplc. because o r :ill- Iiarmonicily, the cncrgy tr;insfcrrcd frorii vibraiion io tr;nisI:ition in the PI,(; trimsition is grcntcr tliini tlic ci i-

ergy transrcrrcd in the PI> collision. Bccausc tlic cncrgy

transferred is greater in the upper level transition, the V- V exchange probability is lower in tlicsc upper IcvcIs. The combined cKcct of increasing matrix clcnicnis ;inti increasing rcsoiiancc dcfcct produce a ni:ixiniuiii in tlic V-V cxchangc probabilities shown. Tlic SSfl rcsiil~s using aV,=aV,.=4.0 A - ' do not agrcc with tlic i i iorc complete Billings and Fisher model. I t is found that Tor T=2000K, using avv=l.47avr for tlic SSH rilles provides very good agreement with tlic scnii-cI;issic;iI rcsuIts. Clearly, either a model niore accurate than SSH iiiiisl hc used for V-V rates or similar compurisoiis niiisl bc performed at a variety of temperatures in oi-dcr to validate any choice of avv. Both of tlicse efforts arc the subject or future work. In this paper, ilic cfrcci or iisiiis various valucs of avv is srudicd.

SSH Rcsults

v cxchangc rates and/or ;I1 low tclll~>cratllrcs, long r:111gc

24 23

41

5

The vibrational transiiion r a t s arc [unciiotis oi' ilic translational temperature arid the population disiribuiion and thus vary in a gas in iliertiial non-cc~uilibriitm. Sonic gcticral features on ilic irmsitiot: rilics can bc cx:iiiiiticd by observing ihc ric:ircst neighbor riiics, K > , , \ , + i , Vor ii

boltzniann distribution. Figure 5 shows the ncarcst ncighhor, V-'1- ir;itisi-

Lion rates for N, at T=2000K. The vibraiional I c \ c I s arc popularcd according to :i bolrzniann dis i r ibui ion t i t

T,,,,=4000K. SSH rates for various values 01 aVI. ntid the rates assumed by Landau-Tcllcr Theory arc s l iow i i . In each or ihc case.s the ircinsiiion r a m incrcxc v i l l i quantum number because of increasing transitioii iiiatrix elements. The SSH raics at the lower levels arc t i c x tlic Landau-Tcllcr r a m . The tipper lcvcl SSH r:iics arc higher than the Landau Teller rates because ~ititi~iriii~iiiic- iiy increases the probabilities of these closcly sp:tcc"d levels relative to the harmonic oscillator Lundnti-Tcllcr model. The lower values of a arc found to givc sonic- what faster rates than the higher values of a. Tlic SSlH raics for av.,.=3.9 A-I arc shown using tlic Kcck iiiitl

Carrier expression and the 3-D numerical fortiiu1:iiioti. There is l i t i lc diffcrcncc in i l i e rcsults of ilicsc two i i ici l i- OdS.

Figure 6 sliows the ticarcst neighbor rates V-7 for CO ?it the same condiiions. As i i i ilic case for ilic h!> - case,

the SSH rates arc higher than tliosc assumcd by ihc Landau-Teller model. Oticc again, ii is foitiid t l i a ilic Kcck arid Carrier expression and ilic 3-D forintilation - pivc very similar resiilts. Because the Kcck atid Carricr formulation requires tiiiicli less CPU iitiic, ii is itscd in all ihc present coniputaiions of [lie vihrcuional sclnaiioii process.

111. NOZZLE F L O W FORMULATIOS

Calcularion OC all of ihc traiisiiion probabiliiics aid rates for one tempcrarure is compuraiionally cxpciisivc. In a general 2-D or 3-D flow we would nccd to coiiiptiic these for every tcnipcraiurc i n the flow field and solve :I vibrational raic cquaiioii for ci icl i vibrational IcvcI. and solve fluid dynamic equations simultaneously. N iirogcn has 57 bound quantum states so wc would nccd io solve over 60 couplcd equations for each grid node. Further. i f a time marching schcmc is used, the CPU rcqitircmcnt would be much grearcr. Thus, detailed calculaiions of ihc vibrational iransition rate equations in general 2-D o r 3- D flows are prohibitively expensive.

Fortunately, vibrational rclaxaiion in cxpandiiig flows has been studicd experimentally in geonicirically simple nozzles. In many of these experiments ilic nozzle a m increases slowly and in the present calculaiions wc make the quasi I-D approximation. This approxitiintion greatly reduces the CPU requirement relative to 2-D and 3-D calcularions because the number of grid notics aiid

cquiltions arc reduced. Thc Ilows siudicd iirc also ;is- sumcd to bc in steady state and inviscid.

Vibraiion;il Raic Eciuet ions 4 The vihraiioiial irotisiiioti raic i i i as~cs cqii;ilioiis arc

o l Ihc form:

. wlicrc N, is i l ic number of i i iolcculcs I in si;itc v per i i t i i i

volunic. Wc can tlicii use fluid iranspori rclaiions to coli- veri [lie iiiaicrial derivative io liniic volume form :itid wriic i l ic quasi I -D, sicady, vibraiioiial iraiisitioii ratc cwatioti as

(111-2)

ti is the strcmiwisc velocity and A is the t iou lc cross scc tioii;il area.

The vibrational cticrzy ~ i c r uni i iii;iss is

111 tlic present calculations. E, and lhits e,,,, arc tiic:isitrcd rclaiivc to thcir ground siiiic (i.c. v=0) viilt~cs. By differ- cntiaiing this equation atid subsiituling i n the prcvious cqiiiiiion wc can show dial ihc quasi 1-D, stcntly, vibrti- iioiial cncrgy rate equation is II

(111.4)

wlicrc k is Bolmiianii's cotisI;uiI atid R =(k/d) is i l ic

ordinary gas cotislaill.

Solution Aleoriihm Although we employed ihc quasi I-D npproxinici-

lion, i f iypical time marching numerical sclicnics ;ire ttscd 10 solve ilic h i i c diffcrcncc Corm of the governing cqon- t io i is iilc calculaiions would still bc coiiipuiniioti~t~ly very expensive. We can take advantage of ilic quasi 1-D for- niiil;iiioti by pcrforming one spatial march rather ihm ii- crating in iinic. For this siudy, a n cl'k%xt, implicii, spncc marching, solver cellcd STIFF7 is utilized. STIFF7 nuiiicrically computes jacohians and iiitcgraics a couplcd sct of quasi-linear, partial diffcrciiiial equations and is dc- scribed i n niorc dcvail by Lomaxi7. An equation is icrined quasi-lincar if the highcsi order derivatives a p pear explicitly and to the firs1 power only. I f Q is a sei of dcpcridcnt variables,

( I I I-S)

then STIFF7 solves a couplcd set of pariial dificrcnrial cauations cncli of the form

x is the indcpcndeni variablc takcn as the strcaiiiwisc co- ordinaic in the present calculations. If wc rake u2 and N,, as dependent variables, then thc vibrational transition rate and energy rate equations shown iibovc f i t thc form rc- quircd for the solver. We may now devclop an cqucilion for u2 by starting from the fluid dynamic cquatioiis Tor steady, quasi I-D, inviscid flow:

puA = constant (111-7.a)

( I 11-7.1))

(111-7 .c.) U2 H + 1 = H, = conshn1t

Assuming uanslational/rotational cquilihrium ;ind 110

chemical reactions gives: H = i RT + e,,,,

whcrc the perfect gas law, p = pRT, lias bccn uscd. DifTcrcnriating and combining each of tlic prcvious c q t w tions yields the required cquarion for u2 wliicli dcscribcs the fluid dynamics of the flow:

(111-8) 1

''

Thus for the detailed calculation of vihrc1lion;il populations coupled to fluid equations the set oi dcpcn- dent variables is

In this formulation, evib and its derivative arc found from equations (111-3) and (111.4). rcspccrivcly.

In converging-diverging nozzlc simiila~ioiis ;I ni i - mcrical difficulty arises because [ l ie mass f low rate for thermal non-equilibrium flow is 1101 known a priori. I f rhc mass flow rate chosen is too high, then a shock will begin to form upstream of the nozzlc throat and sobsonic flow will exist on both sides of the throai. I T the mass f low ratc chosen is too low, tlicn subsonic flow will also exist on each side of the throat. A good discussion of rliis critical mass flow problem and various solutions ;ire

given by Hall and Trcanor.ts In the present calciila~ioiis :in automated trial and error method is uscd wi th the ini- l i a l giicss guidcd b y rhc cquilibriu~n and frozen linii~s. This method is auiomriicd by checking the prcssurc and Mach tiumhcr at each step. I f ;II a given slcp [tic prcssurc is 2% grcatcr rIi;iii ar h e previous srcp, ilicii ;I comprcs- sioti wavc is forming ;itid llic ciilculaiiot~ is rcsrariccl iii ;I slightly lower Mach nuiiibcr b u t with ilic same total prcs- sure and Icnipcriiturc. For locations dowiistrcc~n~ of tlic thro;~t , i f thc Mach number is < 1 whcii A/A* > 1.2 rhcn llic computation is restarted at ;I higher Mach number wirli ilic snmc total conditions. Unlikc other methods, such as pcrturbation methods, the present schcmc prc- scrvcs the exact nozzlc geometry and is simple to implc- nicnt.

Landau-Tcllcr Model Wc can also use the STIFF7 solver to invcstigatc

the performance of the Landau-Tcllcr equation. This is done by omitting the population distribution equaiions and instead of using the true cncrgy rate equation wc iisc the Landau-Tcllcr relaxation cquation. For ;i Landau- Tcllcr solution wc have

Q = [ ' I 2 ) G i b (111-1 I )

wlicrc

and z];, is llic vibrational rclaxation tiiiic. In (his srudy, z~. .~ is obtained from Millikan and White cxperimcntal data. The cquilihrium vibrational cncrgy per i tn i l m;irs for ;I Ihmiioiiic oscillator is

1sothcrni;il Fonnularion The same solver is also used Lo computc vibrational

re1ax;itior o f a constant volume of gas at constanr rransla- tional temperature. Such simulations allow study of many aspects of vibrarional relaxation without additional fluid dynamic complcxitics. lsotliermal simulations arc conductcd by nddiiig the following coiistraints to the rclatioiis described in tlic tircvious sectioi~s: T = constaiit.

d k d u - 0. A = consrant, - = 0, 11 = 1, - - 0. With Llic

dT i1x ii x dx - - _

.~ ~~ ~

velocity fixed ar unity (lie variablc x bccoincs the tinic coord iniilc.

IV. ISOTHERMAL COOLING SIMULATION Rclaxation of a vibrationally excited gas to q u i -

librium a t a lower, constant translational temperature is studied. This siiiiulation is analogous to the vibrational non-cc~uilibriiini ~iroccss i n an expanding flow a11d illus- trates many of the sanic fcaturcs. This simulation is ciisicr to conduct i l i an rhc ~io%%Ic flow calculations bccnusc 110 criiical throa t niiiss f low r i~ tc must bc

7

dctcrniincd for tlic prcscnt isothcrnial, cons tm i volwiic

Considcrablc comput3tion31 i inic can bc s;ivcd i f sonic vcry improbahlc iiiiiltiplc-cliinntitni trimsiiioiis arc neglected. Transition rate caIcii1;iiions for f o r N , :iiitl CO at T<4000K rcvcal vcry sniiill probabilities l o r c d l i s i o n s iti wliicli jumps of grcntcr t h a n two qui in tu i i i IcvcIs arc achieved. Thus in the prcscni isoilicrnial cooling si i i i i i l : i - tion, all 57 bound lcvcls of N, arc computed and only co l - lisional jumps of up to 4 quantum lcvcls arc considered. The required CPU Lime for each of thcsc sii i iul; i i ici i is, is approximately 25 minutes on a Cr:iy YMP.

Initially a volume of N, at 11 = I a tni i s i issuii icd LO

be at an equilibrium bolmiiann distribution ;it 'I\,,l,= 4000 K. Then at I 2 0 the rratisl~itional tcmpcraturc is iiscti ;it T = 2000 K and the gas is allowed to vibratiotially relax. Figure 7 shows the time history of the ground sliilc vibra- tional tcmpcraturc for the Land:tu-Tellcr model and i l ic SSH formulalion with 57 hound vibrational Icvcls and with a,.,=3.9 A-I. The ground statc vibrational Icinpcra- turc characterizes ilic population distribution i n ilic low-

case.

est ICVCIS and is found rrom

SStl results i~rc shown with CI .\,\, = U .\,.,. : i ~ i d w i t h

~.,,=1.47~,.,-. WC scc lliiit cach OC t l ic SSlH C;ISCS slio\vs fastcr rc1;ixatioii 1Ii;in Land:iu-Teller niodcl iii t h i s cooling c;isc. Tlic higher valuc of uvL. rcsults iii l i~s tc r rclaxatioii tlian llic lower value.

The cvolution of the compincd popiiliilioii distr i- butions for tlicsc iwo CRSCS :ire shown on Figir-cs 8 ;ind 9 Tlicsc plots show lhc nuiiibcr density iiornialixcc hy ihc cquilihrium nunlbcr density so that constant dope l i i i c iiiiplics a holtzmann distribution iind a ~.cro slope line ;II unity denotes cquilibrium distrihulion iil the wiiisl:iiioti:il tcmpcrature. The ovcr;ill imprcssion of tlicsc rcxiil is is that over almost tlic ciitirc relaxation proccss. ( t ic lowcr levels distributions arc not far from bolr/.nlann iii sonic vibrational temperature and tlic uppcrmost IcvcIs arc i n equilibrium at T. The ripper lcvcls rciicli ccliiilihriiini vcry quickly bccnusc of the vcry rapid V-T tixisiiioiis for tlicsc Icvcls. Closer inspection of i l ic Iowcr m i triid levels reveals that slopc increases with qii;iiitiiiii nuntbcr which implies that ovcrpopulation exists f o r t l icsc Icvcls. This overpopulation is due to V-V cxcliangc lriiiisitioiis and is more cxtrcnic when Tvib >> T, i.c.tic;ir the

beginning of this cooling proccss. This ovcrI)(II)it13tioii would have been iiiucli tiiorc scvcrc i C i l i c ii1iti;il vibrational temperature wcrc much highct l l i i i n the translational temperature. Even the rcl;ilivcly mild overpopulation for the temperatures shown is s~iCficicnt to accelerate the overall vibrational rcliixation Tlic use of ( ~ . ~ ~ = 1 . 4 7 a , . ~ verses the lowcr value or avv iiicrcascs tlie cflcct of the V-V exchange transilions, ilitis furrlicr accelerating vibrational relaxation and delayin: ihc onsct

01 tlic uppermost lcvcl cquilibrium caused h y V-l iransitions.

Pmniciric Slutly d Because thcrc is uiiccrtainty i n the SStl rues for V-

V cxchangc tratisiiioiis. :I Iiw;iiiictric study was contluctcd involving various viilucs of V-V exchange rates and a,..,. Thc cooling sitnulation was re-computcd using V-V cs- cli;itigc rates that arc a11 multiplied by a factors of .OOI . .01, 0.1, I, 10, 100 and 1000. I n addition, tlic following v;ilucs of avv wcrc studied: c ~ ~ ~ = a ~ ~ . . avv=l.47a,-,., n,3,.=2.0a,.,., and avv=2.5a,.,.. Figure 10 gives the ratio o f ilic Landau-Teller rclaxatioo time to thc overall i~cI. r ~ x c i t t o n . '

solver. The overall relaxation tinic in this figurc is dcfincd ;IS the tinic required for (cvib-euibr~) to Pall to lie of its initial value. Figure 10 shows that with both a\,v=ay.l. and avv=1.47a,.r the overall relaxation rate is approximately 1.4 timcs fastcr ihan tlie rate assumed by the I..andau-Tcllcr model. Wc also see that for these two viilucs or a,,, the overall rclaxation time remains practiciilly constuit cvcn when multiplying or dividing by Pxtors of 1000. This obscrvaiion can bc explained by considering tlic relaxation models proposed by Bray3 and by 'Trcanor Rich and Rchm4. Thcsc models indicalc tha t cvcn without niultiplying tlic V-V exchange rates by I x g c kictors, at the tcnipcraturcs studied in this paper tlic V-V cxchangc r a m arc alrcady much faster than the V-T rates in the lower vibrational Icvcls. The vibralional - Ipopulation quickly atlains a quasi-equilibrium disiribution i n a t ime much shorter than the overall rc- lasaiion tinic. For llows i n which Tvib > T this results i i i

;in ovcrl)ol)itlatii)li which can bc described by :I Trciinor disirihuiion Multiplyin:r t i l l of the. V-V exchange rues b y IO, Cor cs;itiipIc, causes tlic quasi-equilibrium to be re;iclicd I O t imes raster h u t compared to this distribu- tion is already achieved instantaneously. As long as the V-V exchange ground state probability is much greater ilia11 ilic V-T rate (as i t is for moderate and low lempcra- turcs) its exact value has little cffcct on the overall vibra- iioixil relaxation rate. This observation suggests that cvcn i f SS.H Theory significantly under-prcdicts the t r u e iiiagnilude of the V-V cxchongc rates, the overall rclax- iitioii ralc docs not change very much Cot- thcsc types of flows.

Multiplying all of the V-V exchange rates by a coiistant factor greater than unity has little clfcct on the o v c r d l relaxation ratc bul, ;is shown in Figure 10. diffcr- ciit v~iIucs of a,, do affect the overall rate. The scaling of the V-V exchange probabilities relative to Lhc ground .stiilc rate influences Lhe population distribution m d overall rclaxation rate. As shown in Figures 8 and 9, the higher viilucs of avv increase the level of overpopulation in ilie mid and upper Icvcls. Howevcr, Figure 10 dcmon- strates t l i a l only values of avV greater than or equal to -3

than 2.0av.r givc a significantly fastcr overall relaxation

time found from the Cull master equation

8

~ i t c . Nozzle flow cxpcrinicnts of N, and CO ;it tcmpcr:i- turcs from 2000K-4000K obscrvc overall rc1;ixiition rates wliicli arc 3 - 100 tinics faster than tlic Lnndw- Teller value. If thcrc arc sni:ill o r moderate crrors in tlic scaling of tlic V-V exchange r~itcs coniputcd from SSIH Thcoty then tlic overall relaxation ratc is less tIi;iti 2 l imes 1';istcr tliiiti thc Landau-Tcllcr V~IIUC f o r tllc C;ISC considered. Only very significant modifications to h c V- V exchange rates coniputcd from SStl Thcory result i n ovcrall relaxation rates i i iucli faster than 2 for th is ctisc.

"

V. NOZZLE FLOW RESULTS A number of expanding flow cxpcrimcnts have

bccn conducted which quan t i fy the faster vibrational rc- laxation rate in nozzles compared to post-shock Ilows. The test case chosen in this study was pcrfornicd by von Roscnbcrg, Taylor and Tearclg and mcasurcs vibroiional relaxation of pure CO in a 2-D nozzlc. A schematic of tlic iior~lc is shown in Figure 11. In thc cxpcrinicnt a shock is reflected o f f the end wall upstream o f the nozzle. The shock reflection fornis ;I reservoir of high tcmpcraturc, high pressure gas which expands through tlic no%%Ic. This case was chosen bccausc its results arc rcprcscntativc of rhosc cxperiments which had l i t t lc unccrtainty due to impurities, shock diaphragms, and indirect vibrat ional tctiiI)craturc nieasurcnicnt tcchniqucs. The von Rosenbcrg et al. cxpcrinicnt uses an infrared emission tcchtiiquc to nicasurc tlic vibrational tcnipcraturc characterizing rhc population disiributioti i n the lowest Icvcls. The vibrational tcmpcraturc is mcasurcd i n ii i ioulc at a11 xc;i ratio or AIA" = IO for various total tcmpcraturcs and total pressures. The f u l l solvcr is computationally intensive bccausc prohnbilitic.; and transition rates niitst be coniputcd at each step atid 1)ccausc scvcral tries arc nccdcd to dctcriiiinc tlic criticill tliroiit tiiiiss flow rate. Due to limited CPU rcsourccs, the fu l l calculation is pcrfornicd for only he first 57 hound lcvcls of CO. Collisional jumps of u p to 2 quatitimi lcvcls arc considered for this computation. Once the critical mass flow rate is dcrerniincd, tlic n o u l c flow calculrition rcquircd approsimatcly 90 niiiiutcs of CPU time on ii Cray YMP.

Computed and cxpcriiiici<tal rcsiiIts f o r T,=4000K and p,=14.X aim. arc shown in Figures 12 and 13. For tlic SSH rcsults, the ground stiitc vibrational tcnipcriiturc is shown using a,,.,.=4.6 A ~ ' and uvv= 1 .47av.,.. The Landau-Tcllcr calculation predicts relaxation that is iiiucli slower t1i;in the cxpcrinicntal data. The SSH result providcs bcttcr agrccmcnr with the data. Note also that the translational tcmpcraturcs virtually overlay for the Landau-Teller and full solvcr calculations. The c o n - putcd population dismibution for the full solver calcula- tion is shown in Figure 13. From this plot we scc that the vibrational tcmpcraturc for tlic Iowcst lcvcls frcczc bc- fore A/A*=2 in the supcrsonic region. Thc iiiid and up- per levels arc very ovcrpopulatcd and wc scc tliiit popula- tion inversion occurs, i.c. the number density incrcascs

'd

.\J

with quantum numbcr. The full master equation solver with SSH is ablc to prcdict the cxpcrimcntally observed trend but satisPxtory quantitativc agrccmcnt i s no1 nchicvcd.

CONC L US1 ONS I t is shown, herein, that values for tlic intcractioti

range p:iramctcr, U, cati be infcrrcd by comparing tram sition rates to cxpcrinicnval daw In particular, i t is found

CO allow tlic tcmpcraturc dependence of SSH V-T rates to match those rates round from Millikan and White cx- pcrimcntal data. Values Tor thc molecular constants riccdcd in transition ratc theories based u p o n other molccular models may also be inferrcd from a similar comparison mcthod. Thcrc is significant uncertainty in V-V cxchangc transition rates computed from SSH Theory. The crlects of these V-V cxchan, ire rates arc modeled by pcrforming a parametric study of various V- V rates and by using various values of avv. I t is found that higher valucs of avv increase the cfrcct of V-V cx- change rates i n accclerating vibrational relaxation in cooling flows. Through siniulations of a cooling gas and a nozzlc flow i t is found that vibrational relaxation is ac- celerated in conditions or high vibrational cncrgy nnd low tr:uislatiotial temperature. However a parmetric study of V-V exchange ratcs suggests that i t is unlikely tha t tlicsc rates arc solely rcsponsible Cor the cxpcrinicntally observed acceleration ol' vibrational relaxation in expanding Ilows. Ovcrpopulation is sccn in tlic mid ;ind upper levels atid in tlic iio%%lc flow simularion population inversion is predicted. The uccc1cr;itioti nicchanism round Iic'rc agrccs with that proposcd by Trcanor Rich and Rchni and by Bray. The full master equation solvcr with SSH Thcory is able to predict the cxpcrinicntally observed trend but satisfacrory quantirativc agrcenicnt is not achieved. SSH transition rates wi l l undcr-predict the overall relaxation rate only if there are very large crrors in its scaling of the mid and upper levels relative to tlic - mound state rates. Vibrational relaxation should he studied w i t h alternate transition ratc theories to dctcrminc ir they can prcdict the cxperinicnlally ohscrvcd rates. Also, as discussed in rcrerence 2, the lcvcl of impurities in thc experimental gas sample cati have a signilicclnr cffcct on the measured relaxation ratc. Tlic effects of possible impurities in the experimcntal g:~s samples nnd radiative effects must be studied to determine tlic magnitude of the cxpcrimcntal error and to rcsolvc discrcpancics in various cxpcrimenis. Both 01' tlicsc issues must be addressed to allow quantitative agreement between computation and experiment.

t i ~ t a vaiuc or aVr=3.9 A - 1 ior N, and aV,.=4.6 A - 1 for

9

REFERENCES

IMcLarcn, T.I., and Appleton, J.P. " V i b r o l i o n a l Rclaxalion Mcasurcrncnts of Carbon Monoxitlc i n :I

Shock Tube Expansion Wave." Journ;il o r Chci i i ical Physics, Vol. 53, No. 7 , Oct. 1970.

ZHurlc, I .R. , Noncqi~iI ib~-iurn Flows w i i h Slicc.ial Rcfcrcncc to thc Nozzle-Flow Prohlcni," Proccc(1in:s of llic 8th Intcrnational Shock Tube Sympcisiimi. In ipc i ia l Collcgc, London, Jul. 1971.

nBray, K.N.C., "Vibrational Rclaxation of Anharmonic Oscillator Moleculcs: Relaxation iindcr Isoihciiiial Conditions," Journal of Physics B, Proc. Phys. Soc.. Vol. I , Scr. 2, 1968, pp 705-717.

4Trcanor. C.E., Rich, J.W., Rchni, R.G., Vihr;uional Rclaxation o l Anharmonic Oscillators with Euclinngc- Dominated Collisions," Journal of Chcmical Physics. Vol 48, No. 4, Fcb. 1968, pp 1798-1x07

i B l o n i , A.P., Bray, K . N . C . , Prnr i , N . I . I . . ' 'Rapid Vibralional De-excitation Influcnccd by Gasii) ii;imic Coupling," Asironautica Acu, Vol. 15, N o

"Schwar[z. R.N., Slawsky, Z.I., and I-lcizfcltl. k . E , "Calculation of Vibraiional Rclaxation Tinics i i i G.F Journal of Chcniical Physics, Vol. 20, N o 10. OCI. l l ) 5 Z .

'Clarke, J.F., and McClicsncy, Dvnamics 01 R c l i i i i n ~ Gn.;cs, Buitcrworths Publishing Co., M;iss., 1976.

4SIimiia, S.P. , Huo, W.M. , ;ind Park, C.. " 7 W Rare Paramctcrs for Coupled V i b r o t i o n - D i s s o ~ i ~ i i i ~ ~ ~ ~ i n :I Gcncralizcd SSH Approxiiii;ition," AlAA Palxi SS-27 14, J u i i . 1988.

g K c c k , J . , and Carricr, G., "Diffusion Theor!; of Noncquilihrium Dissociation and Rccombiiia~ion," Journal of Chemical Physics, Vol . 43, No. 7, 1065. pp. 2284-2298.

I "Mi I I ikan , R . C., and W h i tc. I). R , , " S y siciii J I I c s o I_ Vibrational Rclasarion," Journal of Chcmic;il I%! sics, Vol. 39, No. 12, Dcc. 1963.

' ILandrurn , D.B. , and Candler, G.V., "Vilmtioti- Dissociation Coupling in Nonequilibrium Flows." :\IAA Paper 91-0446, Jan. 1991.

l*Billings, G.D., and Fislicr, E.R., "VV and \'T R;IK Cocl'ficicnts in N, by a Quantum-Classical Model," Journal of Chcmical Physics, VoI. 43, p p 395-401, 1979.

13Radzig, A.A., and Srnirnov, B.M., Rcfcrcncc Data 011 Atoms, Molcculcs. and Ions, Spi-inger Scr. in Clicinicnl Physics 3 1, Springer-Vcrlng. Bcrlin, 1985

IjLandrum, D.B., private communication, I99 I

"Taylor, R.L., Bitterman, S . , "Survey of Vibrationnl Rclaxniion Data f o r Proccsscs Iinporiant in i l ic C0,-N, d I~ i sc i Sysicm." Reviews o f Modcrn Physics, V o l . 4 I , N o . I , J a i l . 1969, pp. 26-47.

1"Sliarnia. R.D., Brau. C.A.. "Energy Trarislcr i n Ncar- Rcson;inl Molccular Collisions duc LO Long-Range Forccs wiili Application io Transfcr of Vibrational Energy Troin u3 Modc of CO, to N,," Journal or Chemical Physics, Vol. SO, No. 2, Jan. 1969, pp. 924-930.

17Loniax. H. , "Slablc Implicit and Explicit Numcrical Methods Tor Integrating Quasi-Linear Diffcrcntial Equations with Parasilic-Stiff and Parasitic-Saddle Eigcnvalucs," NASA TN D-4703, May. 1968.

IXHaII J.G., and Trcanor, C.E. "Noncquilibriuin Effects in Supcrsonic-Nozzle Flows," CAL 163, Mar. 1968, Corncll Aeronautical Laboratory, Buffalo, NY. Also ACARDograph 124, Dcc. 1967.

l')voii Roscnbcl-g Jr, C.W.. Taylor, and R.L., Tcarc, J.D., "Vilmtional Relaxation o l CO in Noncquilihriurn Nozzle Flow." Jouri ia l of Chcniicd Physics, Vol. 48, 1111.5731- 5733.1968.

u

l ' ~ ' ' l ' ' ' ' l ' r 0.05 0.1 0 0.1 5

T.lI3 [K- l i31 Figurc 3. Comparison of SSH rcsults LO cxperimenlal rcsIIIls for CO,.N, miXtureS , l-llis f igl l rc is reprinted from Taylor and Biltcrnianl5. Scc reference 15 for the sourccs of cxpcrimciilal daui poinls.

Figurc 1. Comparison o f compulcd ground s1aIc V-T transition probabililics from SSI-I ~hcory 10 Millik;ni and While data for N, a i p = 1 atni .

1

1

1 0 0 . . O r i , - 5 0,l

1

1

1

Figure 2. Comparison of computcd ground stale V-T [ransition probabilities from SSH lhcory 10 Millikan and White data for CO at p = 1 ahm.

1 0 ] , : . : : : . . . ~ . ~ , . . . . . . .,. .................................... : : . ,

g .... 0 Billings and Fisher Model .-. SSH with ixvV=eVT

- SSH with ( x v v = l .47u,. 8 . . .

7

OL- 6

- 5 F j 0. 4

3

2 1

- 0 . . . >.- a

. .

. . : : : : / : : : : : / O j ; , : : : : I I I I I i I I I r

0 4 a i z 1 6 20 2 4

vibrational quantum number, v2

Figurc 4. Cornparison of computed V-V probabilities for N, using SSH theory and [ram [lie Billings and Fisher scnii-classical modcl a1 T=2000K with aV,=4.0 A-I.

1 1

..... ............................... ............................. . . . : : : : . :

......... ............... .................. ............. 2 . 0 i : : ............................. ; i... ~ 2 ' 1 4 ~ . .

. . . . . .

. : . . . . . 0.9 ; i I I r I I I

.

r

I I I 0 2 0 4 0 8 0 x 1 oJ

vib energy level [cm. ' ]

Figiirc 6. Conipulcd V-T transitions rates for CO using SSH theory and various values of a. T=2000K and vibrational levels populated in a Boltzmann distrihution at TV,,=4000K.

. , . . ........................ . . I . /t=o.o 'sec . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 os W

z' . 1 o6

1 o2

1 oo

....

............

I , , , \ ' > ( ; , : , . . . . . . . . . . . .

0 5 0 1 0 0 ~ 1 0 ~

vib. energy level [cm"]

Figurc S. Normalized population distribution I'or N, cooling simulation wi th a,.,.=3.9 A-i and Vibrational populalion is normalized by thc cquilibrium distribution.

1 2

v

. . . . . . . . . . . . . . . . . . . . . . . . . . . : , . . . . - . . . .

. . : . . . . . I I I S ' I '

0 5 0 1 0 0 ~ 1 0 ~

vib. energy level [cm"]

Figure 9. Normalizcd populatioii distribution I'or N, cooling siniulatioii with a,.,.=3.9 A - ' and a,,=] .47a,-,-. Vibrational population is noritializcd h y tltc cqiiilibrium distribu[ion.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . " I

. . . . . . . . . . . . I . . . . . . . . . . . . . . . . . . . 1; 1 : .......... ..; . . . . . . . . . i... . . . . . . . . . . . . . . . . . . I , :

............ : .. . . . . . . . . . ............................. 'e

I

............. I v) -0- a" "=(x" ,

..,. : 1 :

1 1 """"" a" "= 2 , 017." , , :

a" "= 2.5U" , 1 ; . . . . . . . . . . . . . . .

............................. ., ........, ~

............................................................ . : ................. I . . . .......

$ 5 . O j : , ; l /I 1 4 , 0 , i:

e

mult ipl icat ive factor for V-V exchange rates

Figurc IO. Overall rclitxatioii t ime for thc cooling simulation using various valucs of R~~ and various niulliplica[ivc [actors for the V-V cnchangc ratcs.

1.33" 8 . 5 . 7 - I

'd t Figure 11. Tcarc two-dimensional nozzlc geometry.

Schematic of von Roscnbcrg, Taylor and

..... ~ ........................................................................................

i I 0 TviTt Exper imental Datal 1 I- Tv/Tt Landau-Teller I

Landau-Teller

.... : ..... * ............ : .....: ......

0 .0 2.5 5.0 7 . 5

x (cm.)

Figiirc 12. Computcd and cxpcrinicntal ground statc vibrational tcmpcrature along the von Roscnbcrg cL. a l . nomic. T,=4000K, pr=14.8 atm., u.,.,.=4.6 A-' and avv=l.47a,,.,..

...............................................................................................

I I I I I I 0 2 0 4 0 6 0 1 0 0 x 1 0'

vib. energy level [cm"]

Figurc 13. Computed normalized popularion distribution fos vo i i Roscnbcrg et. al. nozzlc f low. T,=4000K, p,=14.8 atni., a,-,.=4.6 A- ' and avv=l .47avr.