EXAMPLE SHEET B (Partial Differential Equations) SPRING 2014

Transcription

1 Copuaioal Mechaics Eaples B - David Apsle EXAMPLE SHEET B Parial Differeial Equaios SPRING 0 B. Solve he parial differeial equaio 0 0 o, u u u u B. Classif he followig d -order PDEs as hperbolic, parabolic or ellipic: a T T T b 5 3 c κ u u, κ a cosa d 0 3 η ξ B3. The Crak-Nicolso ie-archig schee for he diffusio equaio κ is Δ Δ κ Δ This ca be rearraged as α α α α α α where Δ κδ α ad, B usig a Vo Neua sabili aalsis ha is, b cosiderig he behaviour of a sigle Fourier ode ik e ω show ha he Crak-Nicolso schee is sable for all ie seps. B. Fid a copuaioal secil for he biharoic operaor o a regular wodiesioal esh, wih equal ad spacig. I wo diesios he biharoic operaor is give b

2 B5. I he -d esh show, values of are sored a discree pois i, separaed b a disace Δ so ha, for eaple, he epressio i- i- i i+ i+ i i i d is a seric secod-order epressio for Δ. d i B eas of Talor-series epasio ad usig also he values i ad i+, fid: i a seric h d -order epressio for ; d ii a seric d d -order epressio for. d i i B6. Ea 008, odified The iegri of solid aerials ca be oiored b easurig he rasissio ad reflecio of soud waves hrough he. Sall logiudial displacees u of a -d ediu saisf he wave equaio u u E ρ * where E is Youg s odulus ad ρ is desi, is he space coordiae ad is ie. a Wrie dow he wave speed c i ers of E ad ρ. b Derive a secod-order, full-eplici discreisaio of Equaio * wih space ad ie iervals Δ ad Δ respecivel. A give aerial has Youg s odulus 50 8 Pa ad desi 000 kg 3. A ie = 0, values of u ad u/ wih esh spacig Δ = 0 are give below u u/ s The subseque evoluio is o be deeried uericall wih iesep Δ = 0.0 s ad side boudar codiios u 0, 0.0, u00, 0 for all. c d e Deerie wheher he chose iesep lies wihi he sable rage. Show how he displacees u afer oe iesep ca be obaied o OΔ accurac usig iiial u ad u/ values, ad calculae hese displacees for he above daa. Use our uerical schee fro par c o calculae subseque displacees a all esh pois up o ie 0. s. Copuaioal Mechaics Eaples B - David Apsle

3 B7. Ea 00 The d -order wave equaio for a depede variable h, is h h c where is ie, is disace ad c is he wave speed. * a b Sae wheher Equaio * is hperbolic, parabolic or ellipic. Derive a secod-order, full-eplici discreisaio of Equaio * wih space ad ie iervals Δ ad Δ respecivel. For waves i shallow waer, h is he deph of waer ad he wave speed is give b c gh where g = 9.8 s is he graviaioal acceleraio. Noe ha c is deph-depede. c Derive he aiu sable iesep for he discreisaio i par b i ers of he esh spacig Δ, graviaioal acceleraio g ad local deph h. Waves are allowed o propagae i a chael wih ed walls. The deph profile a ies = 0 ad = is give below. Mere-secod uis are acil assued hroughou. s The zero-gradie side-boudar codiios h 0 boudar a be approiaed here b seig h0, = h0, ad siilarl a = 60. d Use our discreisaio i par b o deerie he deph profile a =. e Eplai how a iproved zero-gradie side-boudar codiio a be ipleeed uericall b he use of ghos odes. Copuaioal Mechaics Eaples B - 3 David Apsle

4 B8. Ea 007 The eperaure T of a soil laer saisfies he differeial equaio T ct kh * z z where is ie ad z is deph. c = volueric hea capaci ad k h = heral coducivi are cosas. a Sae, wihou proof, wheher Equaio * is ellipic, parabolic or hperbolic, ad eplai wihou aheaical deail wha iplicaios his has for: i he required boudar codiios; ii he pe of soluio ehod. b Se up a eplici fiie-differece ehod for he soluio of Equaio *. For a give soil, c =.0 6 J 3 C ad k h = 0.6 W C. The whole laer of soil is iiiall a eperaure 0 C. A ie = 0 he surface eperaure is suddel reduced o 0 C b a laer of sow ad aiaied a his eperaure. A fiie-differece calculaio wih deph icree Δz = 0. is o be used o calculae he subseque evoluio of eperaure across he soil laer. The eperaure a deph z = a be assued o reai fied a 0 C hroughou. c d e Usig a deph icree of Δz = 0., wha is he aiu sable ie sep Δ ha could be used i our eplici schee of par b? Usig a ie sep Δ = 600 s = 6 hours, calculae he eperaure disribuio i he soil afer hours, usig our eplici uerical schee. Se up, bu do o solve, he Crak-Nicolso schee for he above proble. Copuaioal Mechaics Eaples B - David Apsle

5 B9. Ea 0 Followig a cheical spill he pollua coceraio i he groudwaer, C, varies wih ie ad deph below groud z accordig o he differeial equaio C C κ λc * z where κ is he diffusivi ad λ is a deca cosa. a Sae wheher Equaio * is ellipic, parabolic or hperbolic, jusifig our aswer. b Se up a eplici fiie-differece ehod for he soluio of Equaio *. For a paricular cheical ad groud codiios he diffusivi κ = 0.05 da ad he deca cosa λ = 0. da. Iiiall he coceraio i he groudwaer is zero. Fro he iiiaio of he spill he surface level coceraio C =.0 a z = 0 for all 0, whils a lower ipereable laer a deph z = 5 esures ha C 0 z a ha deph for all ie. c d Usig a deph sep Δz =.0, wha is he heoreical aiu sable ie sep Δ ha could be used i our eplici schee of par b? Usig a deph sep Δz =.0 ad ie sep Δ = das, calculae he cheical coceraio disribuio i he soil afer das usig our eplici uerical schee. Copuaioal Mechaics Eaples B - 5 David Apsle

6 B0. Ea 03 The diffusio of a cheical ijeced io a pipe coaiig sill waer is described b C C κ * where C is he coceraio i kg of cheical per cubic ere of waer, is ie ad is disace alog he pipe. The pipe has legh 3 ad cross-secioal area 0.0. The disribuio of cheical coceraio is o be deeried usig a uifor fiie-differece esh, wih esh spacig Δ = 0.5 ad ed odes coicidig wih he eds of he pipe. Spaial boudar codiios are C/ = 0 a each ed of he pipe. a 0. kg of a cheical is ijeced io fresh waer a disace fro oe ed of he pipe. Assuig ha all he cheical is iiiall associaed wih he ode a his poi, calculae he discree coceraio disribuio i kg of cheical per cubic ere of waer i he pipe iediael afer release. b Se up a eplici fiie-differece ehod for he soluio of equaio *. c d e If he diffusivi κ = 0.05 hr, fid he aiu sable iesep ha could be used wih our uerical schee i par b. Usig a acual iesep of hour, calculae he cheical coceraio i he pipe afer 5 hours usig our eplici uerical schee. Forulae bu do o solve he sei-iplici Crak-Nicolso schee for equaio *, ad sae is advaages ad disadvaages over he eplici schee fro par b. Copuaioal Mechaics Eaples B - 6 David Apsle

7 B. Ea 009 The -diesioal sead-sae eperaure disribuio i a hi eal shee is give b k T ct * where T is he ecess eperaure over he surrouds, k is he coducivi of hea ad c is a hea-loss coefficie. a b Sae wheher his equaio is hperbolic, parabolic or ellipic. Discreise his equaio o a esh wih grid spacigs Δ ad Δ. I a paricular case see Figure a plae of diesios.5.5 has oe shor edge aiaied a a ecess eperaure T = 00 C, whils he reaiig edges are kep a he eperaure of he surroudigs. There are eperaure discoiuiies a he corers. The coducivi is k = 00 W C, whils he hea-loss coefficie is c = 00 W 3 C. T=0 T=00 T=0.5 T=0.5 c Usig equal esh spacigs Δ = Δ = 0.5, fid he ecess eperaure a each of he ieral odes. The hea-loss law is ow chaged o a quadraic versio so ha he eperaure equaio becoes c k T T ** T0 d Use a ieraive ehod of our choice o solve Equaio ** for he sae paraeer values ad geoer as par c, wih addiioal paraeer T 0 = 0 C. Give he odal eperaures o decial place. Copuaioal Mechaics Eaples B - 7 David Apsle

8 B. Ea 003 The sead-sae eperaure disribuio across a -diesioal doai is give b q q 0 where he copoes of he hea-flu vecor hea rasfer per ui area per ui ie are give b T T q k, q k where k is he coducivi ad T is he eperaure above abie. a b c Fid he secod-order parial differeial equaio saisfied b T. Nae his equaio ad sae wheher i is hperbolic, parabolic or ellipic. Wha iplicaios does his classificaio have for is boudar codiios ad for he uerical soluio ehod? A log block of recagular cross-secio Figure A below is ade of aerial wih coducivi of hea k = 00 W K. 3 faces are aiaied a T = 0º C whils he op face has eperaure T = 00º C. Use a appropriae fiiedifferece ehod o he regular esh show i Figure B below o fid he eperaure disribuio over he cross-secio of he block. Use our soluio ad he defiiio of he hea-flu vecor o esiae he oal hea flu per ui legh of block hrough he boo face. Figure A o T=00 C o T=0 C o T=0 C 0.3 o T=0 C 0.8 Figure B Copuaioal Mechaics Eaples B - 8 David Apsle

9 B3. Copuaioal eercise The d -order wave equaio for waer deph h, i a chael is h h c where is ie, is disace ad c is he wave speed. a b Derive a secod-order, full-eplici discreisaio of his equaio wih space ad ie iervals Δ ad Δ respecivel. If he ode separaio is Δ = 0 ad he wave speed c = 3.0 s, fid he aiu sable iesep for our discreisaio i par a. Waves are allowed o propagae i a chael wih ed walls. Waer deph h ad is ie derivaive h/ are give a ie = 0 i he able below h h s c d e Show how he deph h afer oe iesep ca be obaied o secod-order accurac usig iiial h ad h/ values, ad calculae hese dephs for he above daa usig a iesep Δ = s. Eplai how a zero-gradie side-boudar codiio h 0 boudar a be ipleeed uericall b he use of ghos odes. This boudar codiio is o be applied a boh eds of he chael i his proble. Use our discreisaio fro above, wih a iesep Δ = s, o calculae he waer dephs i he chael a ies up o = 0 s. Plo he waer deph profiles a ies = 0, 5, 0, 5, 0 s o a sigle graph o show he ie evoluio of he profile. You a use a copuaioal ool ha ou like o help ou. Copuaioal Mechaics Eaples B - 9 David Apsle

10 B. Copuaioal eercise The Figure below is a fiie-differece represeaio of he cross-secio of a square bar subjeced o pure orsio. h The sress fucio is foud o be disribued across he bar accordig o a Poisso equaio 0 wih = 0 o all boudaries. a Se up he discreised equaios for a esh size Δ Δ h. b c d e Solve he discreised equaios for he esh show wih h =, b a direc ehod e.g. Gaussia eliiaio. Noe: his is ade a lo easier if ou observe he geoeric series of he proble. Se ou he Gauss-Seidel ieraive schee for he proble. Sar wih a iiial value of a all ieral odes. Carr ou a leas 3 ieraios for he whole esh i.e. igore he series of he proble. Repea c wih he SOR successive over-relaaio ehod wih a over-relaaio paraeer ω =.5. Wrie a progra i a laguage of our choice o do he work i c ad d. Copuaioal Mechaics Eaples B - 0 David Apsle