Characteristic Equations and Boundary Conditions

Transcription

1 Charatriti Equation and Boundary Condition Øytin Li-Svndn, Viggo H. Hantn, & Andrw MMurry Novmbr 4, Introdution On of th mot diffiult problm on i onfrontd with In numrial modlling oftn li in tting th boundary ondition; whr th modld domain intrt with th outid univr. In th following w will onidr on of th mot lgant mthod for olving thi problm. Th modl quation that i appropriat to thi diuion i th advtion quation + u = whr i.g. th dnity and u th fluid vloity. Th olution to thi quation i wll known: i imply advtd with th flow. How do w would trat thi quation at a boundary? Not that all information i tranportd propagat through th trm, o tting th boundary ondition i quivalnt to pifying at th boundary. In gnral w will try to rform our quation into a form uh that w an onidr thm a advtion quation; i.. if our quation an b writtn in th form U + λ U = w know that th quantity U i advtd with th pd λ. At th lft boundary = a th tratmnt of th quation dpnd on th ign of λ λ > impli that th fluid i inflowing and what i to imping on th modlld ytm. mut b givn by additional xprion pifying λ < man that th fluid i outflowing in whih a w nd to allow th known olution to propagat out. Th drivativ i givn by th intrnal olution,.g. by omputing a on idd drivativ on th grid point to th right of = a. At th right boundary = b th tratmnt of th boundary for thi quation i xatly th onvr, whr now λ < indiat inflow and λ > outflow. Not that vn whn w hav our Q or ink trm L in our modl quation U + λ U = Q L th am prription an b ud to t th boundary a long a Q L do not dpnd of th patial drivativ of th dpndnt variabl u,,.... Thi quation giv an a xampl of a haratriti: A wav mod propagating in a givn pifid dirtion and vloity λ. If all our dpndnt varibabl U i an b obtaind from quation lik U i + λ U i i + C i =

2 whr C ha no dpndn on any Uj, boundary ondition would b impl to driv; i.. find th ign λ i at th boundary and thn ithr pify th patial drivativ from th inflowing information or th outflowing intrnal tat. Our aim mut thrfor b to find out what wav mod thr ar and thn trat ah wav mod paratly, dpnding on th ign of λ i at th boundary. Lt u b onrt and onidr th D quation of ma and momntum onrvation. In that a w ar an writ our ytm of quation a + u + u = u + u u + p g =. W annot immdiatly writ th quation in th dird form in thy both ontain our/ink trm that ar funtion of patial drivativ of a dpndnt variabl u and p whih i a funtion of through th quation of tat: p = p. Bau of th xtra drivativ th wav do not travl at th vloity u, and th quation ar oupld. Thrfor w annot pik out a partiular advtion vloity; in fat it i a mixtur of wav mod travlling in oppoity dirtion. So how do w handl u p and u at th boundary? A it tand w don t konw if if w hav inflow or outflow and in fat w hav both. In ordr to trat th boundari with th mthod of haratriti w will tak a ombination of and u and form nw quation that do travl at wll dfind vloiti. Aftr all w an ombin quation for and u in any way w want add, multiply,... Th linar algbra mahinry Fluid quation may alway b writtn in onrvativ form, in thy xpr th onrvation of partil, mattr, momntum, nrgy, t. or in gnral w writ whr Ũ = u u + uu u + p + u + u = + p g = or 3 g = 4 Ũ + F + D = 5 F = u p + u D = g Lt u now introdu th primitiv variabl U, whih w may hoo a w lik,.g. U =. 6 u Thu by th hain rul U i = Ũi U j U j whr w dfin P ij Ũi U j

3 Similarly w hav for th flux F i = F i U j U j whr w dfin Q ij F i U j Thu w an rwrit th onrvativ form of th quation 5 a P U + Q U + D = whih w immdiatly multiply by P and dfin A P Q and C P D to giv u th primitiv form of th quation U + A U + C = 7 Thi quation look almot lik th advtion quation quation w ar king; if only th matrix A had bn diagonal, i.. if only that A ij = λ i δ ij, whih, for th hydrodynami quation and th primitiv variabl 6 A maniftly i not: Computing A via th omponnt of P and Q from th dfinition P = Ũ U =, P = Ũ U u and thrfor that P = u P = u/ / u A = P Q = p u Q = p + u = u, t w find that u So, o far w not don anything at all rally, but lt u tak advantag of th fat that w may tranform th quation in any way w lik, i.. w want to diagonali A uh that A r i = λ i r i whr λ i, r i ar igvalu and right ignvtor of A rptivly. W will alo nd th lft ignvtor of A: l T i A = λ i l T i Right multiplying thi xprion with r j w find that r i and l T j ar orthogonal: l T i A r j = λ i l T i r j l T i λ j r j = λ i l T i r j λ j λ i l T i r j = l T i r j = if λ j λ i In omponnt form thi i writtn A nk r ki = λ i r ni. W now dfin th matrix S uh that it olumn ar ompod of th right ignvtor r i, i.. S ij r ij, whih man that w an writ A S = S Λ whr Λ ij = λ i δ ij but thn, by right multiplying thi xprion by th invr right ignvtor matrix S w find Λ = S AS and S A = ΛS 3

4 Clarly w mut hav that l ji = S ij f, whr f i a normaliation fator that nur that S S = I. Lt u now rturn to quation 7, th primitiv form of our quation, whr w lft multiply by S. Thi giv U S + S A U + S C = 8 S U but in S ij = f l ji w hav in omponnt form S U j ij f l ji + Λ S U + S C = 9 + Λ iks U j kj + S ij C j = U j + λ is U j ij + S ij C j = or, in vtor form again U f lt i + λ U i f lt i + f lt i C = now imply multiply by f and w finally arriv at th haratriti quation l T i U + λ U i + lt i C = 3 Exampl: D adiabati or iothrmal flow Lt u now driv boundary ondition for D adiabati or iothrmal flow, howing at th am tim th gnral mthod of driving boundary ondition for any problm with ral ignvalu. W hav that A r = λr = λi r A λi r = A non-trivial olution i found if dta λi = or for our a u λ A = P Q = p u λ Rmmbring that = p w that th dtrminant yild u λ = or λ = u ± W thrfor find two ignvalu whih aording to onvntion w ordr from th mallt valu to th hight,.g. in gnral λ λ... λ n ; o λ = u and λ = u +. W an now u th A matrix and our nwly drivd ignvalu to driv right and lft ignvtor: r = r / = l T / = / l T = / Armd with th lft ignvtor w ar now in a poition to u th gnral haratriti quation on our pial a: + u u + u u + g = + u + + u + u + u g = 4

5 Unfortunatly, in =, t w annot writ th quation in th form of tru advtion quation for th quantiti +u/ and u, but nvr th l th quation do rprnt wav/diturban propagating at th haratriti pd u + and u rptivly. Sin th dtrmination of boundary ondition i all about handling th patial drivativ lt u writ λ i l T i U L whih allow u to rformulat th primitiv form of th quation givn by 8 by lft multiplying by S U + S L + C = If w prfr to hav th quation in onrvativ form w mut lft multiply by P, thi giv intad Ũ + P S L + D = Noti that a ombination of L will our in ah of our quation. In th pifi a of our D adibati hydrodynami quation w find + L + L = u + L L g = whih larly how how wav travling in th poitiv u + L and ngativ u L dirtion ar ontaind in both th ontinuity and momntum quation. 3. Boundary prription Look now at th right boundary = b, in thi a for uboni flow λ = u < rprnt inflow and λ = u + > rprnt outflow. Thu w nd to omput L from om pifid boundary ondition, whil L hould b omputd bad on th intrior olution. 3.. Flow into a olid wall In thi a w rquir u =b =, thu alo u =b=. Th momntum quation now giv L L g = and thrfor L = L g Th ontinuity quation now giv u th rquirmnt for th tim drivativ of + L g = whr L i omputd by ontruting a on-idd drivativ from th intrior point. W hav found th fator f = by rquiring S S = I and rmmbring that S = /fl T. 5

6 3.. Suproni outflow In thi a w aum that u > o that both λ > and λ >. Thi man that both haratriti ar outflowing and hould both b omputd by xtrapolating th intrior olution to th boundary Suboni outflow Hr w hav th am ituation a in th a of flow to a olid wall in that L om from a pifid boundary ondition whil L i omputd bad on intrior point. Of pial intrt i to t up a boundary ondition for a non-rflting boundary ondition. W thn rquir that th amplitud of th inoming wav hould b ontant in tim i.. thr hould b no wav. Th inoming wav i dfind by u + u whr L = u u. W rquir that Hn, th quation at th = b boundary ar whr, again, L ar t by th intrior point. u + g = u = L = g + L g = u + L g = 4 Charatriti quation for MHD Th quation w ar olving ar inluding B trm: u + u = + uu + P µ B B ɛ µ BB = g B Writing th in trm of th tim and -drivativ: u h + u + P u = Q 3 u B + u B =. 4 + u + u = u H H H u H 5 u h + u B B h µ ɛ B h B µ = u H H u h hp + µ B H H B h ɛ µ B h H B H µ hb + g h 6 6

7 u + u u + P + B H µ B H ɛ B B µ = u H H u + µ B H H B + ɛ µ B H B H + g 7 B H + u + + P u = u H H + P H u H + Q 8 + u B H + B u H B u H = u H H B H B H H u H + B H H u H 9 B B + u = u H H B B H u H + B H H u Th A matrix i thrfor u u B µ ɛ Bx µ u B µ ɛ By µ A P = u P µ µ ɛ B µ + P u B u B u u Th ignvalu λ of A ar givn by u λ u λ B µ ɛ Bx µ u λ B µ ɛ By µ P u λ P µ µ ɛ B µ + P u λ B u λ B u λ u λ λ u λ u B µ λ u λ u B µ λ u B µ = = 3 W dfin vloiti, = a = B µ = B µ P + + P P = a + + a + 7 7

8 = a + a If B and B H, thn th rulting ignvalu ar Λ = u, u, u +, u, u + +, u +, u +, u. 3 If B and B H =, thn a =, + = and =, and o th rulting ignvalu ar Λ v = u, u, u +, u, u +, u, u +, u. 3 If B = and B H, thn =, + = a + = o and =, and o th rulting ignvalu ar Λ h = u, u, u, u, u + o, u o, u, u. 3 8

9 Eignvtor for λ = u P w 4 = 33 w 6 B = 34 w 7 B = 35 w + w 5 + P + w 6 + w 7 = 36 P w 4 = 37 B w µ + w 4 µ B w 3 µ + w 4 µ B ɛw µ ɛw 3 µ ɛw 4 µ = 38 = 39 = 4 Thrfor w 4 =, w 6 = and w 7 = from whih follow w = and w 3 =. w 8 i unpifid and w and w 5 ar rlatd by w ++P w 5 =. Thr ar two unknown lft, produing two dgnrat ignvtor. I hoo: l =,,,,,,,, 4 l = + P,,,,,,, 4 9

10 Eignvtor for λ = u + P w 4 = w 43 w 6 B = w 44 w 7 B = w 3 45 w + w 5 + P + w 6 + w 7 = w 4 46 P w 4 = w 5 47 B w µ + w 4 µ B w 3 µ + w 4 µ B ɛw µ ɛw 3 µ ɛw 4 µ = w 6 48 = w 7 49 = w 8 5 If B w = w 5 = P P w 4 5 w 4 5 w = B w 6 53 w 3 = B w 7 54 and o w gt th quation: w 4 + w 6 + w 7 = 55 w 6 + µ w 4 = 56 w 7 + µ w 4 = 57 ɛ B µ w 6 + ɛ B µ w 7 ɛ B µ w 4 = w If = and B H thn from quation 56 w 4 = and w 6 + w 7 =. So that w 8 = and l 3 =, ignb, ignb,,,,, B H B H µ B H µ B H 6 If = ± thn w 6 = Bx µ w 4, 6

11 w 7 = By µ w 4, 6 w 8 = ɛ a B µ w o that [ w 4 B x µ + µ ] = 65 and ± atify thi quation for any w 4. l 5 = l 7 = P P, +B +, B µ, +B + µ, B µ, +, P µ,, P,, + + µ, + + µ, µ, µ,, 66, 67 68

12 Eignvtor for λ = u P w 4 = w 69 w 6 B = w 7 w 7 B = w 3 7 w + w 5 + P + w 6 + w 7 = w 4 7 P w 4 = w 5 73 B w µ + w 4 µ B w 3 µ + w 4 µ B ɛw µ ɛw 3 µ ɛw 4 µ = w 6 74 = w 7 75 = w 8 76 Obviouly w 8 =, and if B w = w 5 = P P w 4 77 w 4 78 w = B w 6 79 w 3 = B w 7 8 and o w gt th quation: w 4 w 6 w 7 = 8 w 6 µ w 4 = 8 w 7 µ w 4 = 83 ɛ B µ w 6 + ɛ B µ w 7 + ɛ B µ w 4 = w If = and B H thn from quation 8 w 4 = and w 6 + w 7 =. So that l 4 =, ignb, ignb,,,,, B H B H µ B H µ B H 86 If = ± thn w 6 = Bx µ w 4, 87

13 w 7 = By µ w 4, 88 w 8 = ɛ a B µ w o that [ w 4 B x µ + µ ] = 9 and ± atify thi quation for any w 4. l 6 = l 8 = P P,, + B + B µ, + B + µ, B µ, +, P µ,, P,, + + µ, + + µ, µ, µ,, 9,

14 Th matrix S i a matrix ontaining th ignvtor l: + P ignb By B h ignb Bx B B h y µ B h µ B h ignb By B h ignb Bx B B h y µ B h µ B h S = P B + µ B + + µ P µ + µ + P B + B + µ + µ P µ + µ + P B µ B µ P µ µ P B µ B µ P µ µ Many of th valu ontain diviion by ro in om irumtan if alulatd traightforwardly. Thy an b rwrittn a: + P R R y R x y µ Rx µ R R y R x y µ Rx µ S α = + P α R x α R y α + α + P R x α R + µ y α µ 96 α + P α R x α R y α + α + P R x α R + µ y α µ α P α R x + α + R y + α + α P Rxα+ µ Ryα+ µ α P α R x + α + R y + α + α P Rxα+ µ Ryα+ µ whr 95 = ignb 97 R x = B h 98 R y = B h 99 α + = + α = + + V Uing ΛS = d w thn find d givn by B d = u d = u + P d 3 = u + R y u x + R x u y + R y µ R x µ

15 u x d 4 = u R y α+ P d 5 = u R x α µ α+ P d 6 = u + u y R x + R y µ R x µ u x R x α u y R y α + u +α + + R y α µ + u x R x α + u y R y α u +α + + R x α µ + R y α µ α P d 7 = u + R x α + µ α P d 8 = u R x α + µ + u x R x + α + + u y R y + α + + u α R y α + µ u x R x + α + u y R y + α + u α R y α + µ Th invr of thi matrix, S i: P α + α + α α R y R y α R α x R +α + x R x +α + R x R x R x α R α y R +α + y R y +α + R y +α + +α + α S = α P +P +P +P +P α + α + α α µ µ µ α R y R µ α y R x R x µ α + R x µ α + R x µ R x µ µ α R µ α x R y R y µ α + R y µ α + R y Thi vrion bhav wll undr all irumtan a long a > if R x, R y, α + and α bhav wll. R x and R y ar only a problm whn B h = Bx + By, but in thi a and ar alo and th limit i finit. α + and α only hav a problm whn a =, but th limit i alo finit hr alo. If h = δ in θ and = + δ o θ whr δ thn α + o θ α + o θ δ + δ o θ + δ + δ o θ Th boundary quation in trm of th primitiv variabl ar thn givn by 3 = [ ] P d + α + d 5 + d 6 + α d 7 + d 8 5

16 u x u y u B = = u H H H u H 4 [ R y d 3 + d 4 + α R x d 5 + d 6 + ] +α + R x d 7 d 8 P + B µ u H H u x + x µ B H H 5 [ R x d 3 d 4 + ] α P + B µ y R y d 5 + d 6 + +α + R y d 7 d 8 u H H u y + µ B H H 6 = [ + α + d 5 d 6 + α d 7 + d 8 ] u H H u + g + µ B H H B 7 = [ P + + P α +d5 + α + d6 + ] + P α d 7 + d 8 H u H P H u H + Q 8 [ µ = R y d 3 + d 4 + α R x d 5 + d 6 α ] + R x d 7 + d 8 u H H + H u H + B H H u x 9 [ µ = R x d 3 + d 4 + α R y d 5 + d 6 α ] + R y d 7 + d 8 u H H + H u H + B H H u y = u H H B B H u H + B H H u 6

17 For atual boundary ondition, th d ar hangd for inoming haratriti. Th way thi i don in th urrnt od i dribd hr. Th quation an b writtn: V + Sd = C V S = d 3 whr V i a vtor ontaining th variabl, u, and B. d i a vtor ontaining d d 8, and C i th trm of th original quation not ontaining any -drivativ. Th quation in trm of th haratriti variabl an b writtn: V S + d = S C. 4 Th boundary ondition for th inoming haratriti mut atify th tati a whr u = and V =, o that d i = S C u=. 5 C u= i for th magnti fild quation, and th dnity quation. For th othr quation: C u= = Q 6 C u u= x = P + B + x µ µ B h h 7 C u u= y = P + B + y µ µ B h h 8 C u u= = g + µ B h h B 9 Auming Q =, thn th inoming d an b alulatd to b: d i = 3 d i = 3 d i 3 = [ R h h P + B + B ] H h B h 3 µ µ d i 4 = d i 3 33 d i 5 = + α + g + µ B h h B + α R h h P + B 34 µ d i 6 = d i 5 d i 7 = α g + µ B h h B + α + R h h P + B µ d i 8 = d i

18 For xampl, from quation 6, 5 and 6 to 9, w gt: [ d i 3 = R y P + B + ] x µ µ B h h [ + R x P + B + ] y µ µ B h h = R y x + R x y P + B µ + µ B h x + B y x B x y B x y + µ B h x B y y + B x x + B x y = R h h P + B µ By x y + Bx + B µ B y h [ Bh h B h R h h µ = ] P + B µ