An Analytical Calculation of Strong Shock Wave for Frozen Compressible Gas Flow Produced By Plane Piston

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Leslie Rice
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1 Advace i Mahemaic ad aiical ciece A Aalyical Calclaio of rog hock Wave for Froze Comreible Ga Flow rodced By lae io Kamyar Maor Dearme of Aeroace Egieerig Amir Kabir iveriy of echology ehra Ira 8-44 ad Flow eearch ad Egieerig 88 Malewood Avee alo Alo CA 94 maor@a.ac.ir Abrac: he ime erie eaio of he Eler eqaio for he eady oe dimeioal ga flow of a ivicid comreible erfec froze ad diaomic ga rodced by lae io are obaied o erm eacly by mea of ymbolic calclaio. he aim i o m he erie ad for ha ade aroima ha bee ed i origial form o eable o icreae he rage of alicabiliy of he erie. By doig o we are ccefl o icreae he rage of he validiy of he erie ad hee are eac calclaio o arbirarily figre of accracy. I oher word we have olved he oliear arial differeial eqaio eacly by mea of comer ad ha i a real cce. Key-Word: Comreible Noliear Eqaio ymbolic Calclaio Irodcio he ime erie eaio of he Eler eqaio for he eady oe dimeioal ga flow of a ivicid comreible erfec diaomic ga rodced by lae io are obaied o 6 erm eacly by mea of ymbolic calclaio ad comared wih or merical calclaio. I eem ha he erie ha radm ig aer.ha mea he eare iglariy i omewhere i comle lae of ime. he aim i o m he erie ad for ha ade aroima ha bee ed i origial form o eable o icreae he rage of alicabiliy of he erie. For imilar roblem of oeial flow a circle here i wo hree ad dimeio [] we have bee able o deermie he locaio of iglariie. ade aroima ha bee ed i origial form o eable o icreae he rage of alicabiliy of he erie a ha bee ed i he work of Maor [][] ad adyke [][4]. We ado a aleraive aroach o he global roblem which coi i olvig he iiial-vale roblem by eadig he olio i ower of ime. hi mehod wa irodced by Lee [] for a cylidrical io ha collae wih eed roorioal o a ower of ime. He calclaed hree erm of a doble eaio i ower of boh ime ad ad a arameer rereeig he dearre of he hock wave from ifiie regh. Laer Bach & Lee [6] carried he ame kid of calclaio o for erm for boh cylidrical ad herical wave ig he more comlicaed iiial codiio ha he flow i rodced by he iaaeo deoiio of eergy a a fiie radi. he fir aroimaio i he he elf-imilar olio for a rog laar bla wave. We coider a erfec ga a re i a emi-ifiie raigh be of coa cro ecio ad boded o oe ide by a io. he io i e i moio io he ga a ime = ad i oiio i oed kow a all laer ime amely. ico effec are egleced i he coeqe moio of he ga which i oed o be oe dimeioal deoig by he diace alog he be we ca ake he ga o occy he regio whe = ad whe > where ad recribed. If he velociy dow he be he rere deiy ad emerare of he ga a are deoed by reecively ad i he raio of he ecific hea of he ga. Afer ome dy o abiliie of differe choice of ier cell fle for he roblem. he aov fl wa choe which eemed o be he mo efficie ad rob merical mehod i he reece of very owerfl hock a we have here. Eqaio he eqaio of coiiy momem ad eergy are i [] edov IBN:

2 Here i he velociy are deiy ad rere reecively. he bodary codiio are : a he ffi deoig give coa vale ad a From he give moio of he io I fac of core he whole ga i o i moio a ay fiie imeb he bodary of he movig ga beig a hock wave ad if codiio j behid he hock are he deoed by a ffi he he aroriae jm codiio acro he hock are : A ifiie Mach mber ad fied. he akie-hgoio relaio give he codiio j behid he hock wave a referece []edov 99.. or he baic aroimaio for mall ime i j he flow rodced by imlive moio of a lae io; ad we ame ha he eed i o grea ha he Mach mber of he hock wave i effecively ifiie i ia ermiology we eglec he coer -rere or make he cold-ga aroimaio o hawe eed oly a igle erie. By delegaig he moig arihmeic o a comer we have. i a few mie calclaed6 erm of he eaio i ower of he ime. here by aalyzig he coefficie we are able o erac wih good accracy he iglar local rcre of he flow. eddal [8] aemed o olve he ree roblem by coiderig he lae io whoe oiio afer ime = i ad eadig he olio i ower of. A mall ime he flow i aroimaely ha rodced by a lae io movig io qiece ga wih eed I rodce a hock wave movig a coa eed. Bewee he io ad he hock wave he ga ha coa eed = deiy ad rere. We irodce dimeiole variable ig a referece legh a eed ad a deiy eig: / ; ; ad he immediaely droig he bar he we ge he ame eqaio a bjec o bodary codiio. 4 a a he or baic olio for he lae io i: o ake advaage of he coical are of hi baic flow We relace by a variable ha varie from zero a he io o iy a he baic oiio of he hock wave. he he differeial eqaio become : Advace i Mahemaic ad aiical ciece IBN:

3 8 6 Ad he bodary codiio 4 are 9 a a... erie Derivaio ad Comer Eeio I eem reaoable o ame ha he olio i aalyic i ime. We herefore ead he kow oiio of he hock wave i a aylor erie a: Ad likewie ead he flow variable a: 4 Here or baic olio give biig he above eaio 4 io he differeial eqaio 689 ad eqaig like ower of yield a eqece of riad of fir-order liear ordiary differeial eqaio for he coefficie ' which for he ecod aroimaio are:...; 4...; 4 ;... 4 For imliciy we give he rel oly for For Advace i Mahemaic ad aiical ciece IBN:

4 Advace i Mahemaic ad aiical ciece for he oiio of he hock wave for Whe we form he raio [/] [/] of heade aroima he rage of alicabiliy of he erie icreae. Moreover whe we form he raio [/] [/] ad [/] of he ade aroima for he Figre ca be corced from hee hree raio a ca bee ee hey agreed o he vale = for he oiio of he hock wave for.4.. Figre. ariaio of he oiio of hock wih ime i... i. Aalyi of erie ad Dicio ade aroima ha bee ed i origial form o eable o icreae he rage of alicabiliy of he erie a ha bee ed i he work of Maor [] ad Maor []. hi mehod doe o ecearily reqire ay iformaio abo he radi of covergece. he ade aroima rovide a aroimaio ha i ivaria der a Eler raformaio of he ideede variable. he heory of ade aroima ha bee ed eeively i Maor []. Briefly aed he ade aroima i he raio /Q of olyomial ad Q of degree m ad reecively ha whe eaded agree wih he give erie hrogh erm of degree m+ ad ormalized by Q =. ch raioal fracio are kow o have remarkable roerie of aalyic coiaio. he coefficie of he ower erie m be kow o degree m+. By eqaig like ower of g ad /Q he liear yem of m++ eqaio m be olved o obai he coefficie i he fcioal form /Q ade aroimaio of order [/] [/] ad [/4] for average emerare gradie erie are reecively: ade[/]: Coclio he ime erie eaio of he Eler eqaio for he eady oe dimeioal ga flow of a ivicid comreible erfec diaomic ga rodced by lae io are obaied o 6 erm eacly by mea of ymbolic calclaio ad comared wih merical. Afer ome dy o abiliie of differe choice of ier cell fle for he roblem. he aov fl wa choe ef[] which eemed o be he mo efficie ad rob merical mehod i he reece of very owerfl hock a we have here.he we ried o comare he aalyic coiaio by ig ade aroimaio o or merical mehod of aov. By doig o by ig [/] ade wih he merical mehod a we icreae Mach mber by redcig he merical calclaio aroache o or erie olio which i doe for Mach mber ifiiy. eferece: []. K. Maor H. Moradi. Mazraeo rog hock wave for comreible ga flow rodced by lae io roceedig of he Fir Ieraioal Coferece o Mahemaic ad aiic harjah.a.e.. 8- March. [] K. Maor ig erie Eeio Mehod For olvig boic Flow a A Circle roceedig of he Eigh Cogre of Flid Mechaic hezhe Chia.. 6- December 999. [] M. D. adyke A.J.Gma he Covergig hock Wave from a herical or IBN:

5 Advace i Mahemaic ad aiical ciece cylidrical io. J. ' Flid Mech. 98. [4] M.D.a Dyke A. J. Gma boic oeial Flow a a circle ad he raoic corovery J. Aral. Mah.oc B [] G. G. BACH J. H. "LeeIiial roagaio of imlively geeraed covergig cylidrical ad herical hock wave" J. of. Flid Mech [6] B. H. K. Lee "he iiial hae of collae of a imlodig hock ad he alicaio o hyeroic ieral flow"c.a..i. ra [] L. I. edov "imilariy ad Dimeioal Mehod i Mechaic" Academic re 99. [8] W. F. eddall "he aymoic rajecory of a rog laar hock wave ariig from a o-elf-imilar io moio" h.d.dieraio aford iv. aford Calif. 9. [9] K. ewaro B.W. homoo "oedimeioal eady flow a ifiie Mach mber" roc. oy. oc. A [] K. Maor Lamiar Flow hrogh a lowly oaig raigh ie J Flid Mech [] K. Maor ig oke Eaio For Naral Covecio Iide A wo- Dimeioal Caviy Flid Dyamic eearch [] G. A. Baker he heory ad alicaio of ade aroima mehod {I Advace i heoreical hyic ed. K.A. Brecker} vol [].. aov "Calclaio of Ieracio of No eady hock Wave wih Obacle" J. Com. Mah. hy vol IBN:

Lab 2 Position and Velocity

Lab 2 Position and Velocity b Lab 2 Posiion and Velociy Wha You Need To Know: Working Wih Slope In las week s lab you deal wih a lo of graphing ideas. You will coninue o use one of hese ideas in his week s lab, namely slope. Howeer,