可壓縮性多相位流與非凸性狀態方程的數值解法(1/2)

全文

(1)

(2)

(3) ! "# %$& # &'(*)+-, ./ 0 123 4 65798"1: ;

(4) =<9 #8?>@BA CEDGFHJILKNMPO(Q#F;R#S@DT U6S@MPD(VXW@Y[Z6\]\6^ _*`badcfehgiakjl

(5) mbnoij"gijqprnsgtoucpbg?jJvkakwakmhxgiyzg?jqci{bv|oucl}nsg~o?`bj"mG{hwjqgiadJxvctoi{hbynl

(6) mbnsmbvakmbjxgmbnmhJvkxsciciadJxv x jqcl}ng*jJmhcijsxc[hn cq#j{rctj-oi`bj4[{bvkjJgjqG{hxoiaknmhcnl0GxcyXmhxwadJcxcx~wnXbjJv;ctyctoijqwBNl; } xmrouxjo?`bjprnspb{bvdxg xmEjqgxsxvdc*jqG{hxoiaknm"nl@cto?xoijnloi`hj4l}ng?w" [? [¡£¢ ¤¦¥¨¥©

(7) § ªL« *¬+3 ® ¥ ~ ¯ xc@xwnjqvGl}ng#oi`bjoi`hjJg?wnyXmhxwakerjq`hx Xansg#nlhgijxvGsxsctjc±°²³

(8) jJg?j jqmbno?jqcoi`bj*pbg?jqc?ci{bgijs/jqmbno?jqc oi`hj~ctp6jqJa|´6 nsv{bwjsh ~jqmbno?jqc

(9) oi`bj~cip6jqa´r4akmGoijJg?mhxv0jJmbjqgisyhxmh © 6xmh ¤ bjJmbnoijqcoi`bjwnsvj{bvdxg ns`bjqciak jfl}ngujqcq;´hmha|o?j¦ctakµJjnlo?`bj¶wnvkjq{hvjcJxmh·oi`bj¶guxo?an"nlciprja´r`bjqxo?cq0g?jqciprjoiak jqvys¸bnsg4oi`bj oyXprjnl0pbg?nehvjqwcnsmhctadjqgij`bjJg?jGa|oakc jqgiy{hcijl}{bvroinj´rmbj

(10) o?`bjcin¹ºJxvvkjqfl}{bmrbxwjJmGouxv]jJg?a xoiak j

(11) nl sxsc*yXmhxwadJc»¼xcl}nvkvkn cJ½ »¾¡ ¥ ¿ÀNÁ ÄÁ ¯ ]]ÂÂ Á;Á; ¯ Ã iÃ xmr{rctjaoxcx xy¶o?njo?jJg?wakmbj `bjJoi`bjqgxfh{badEcto?xo?jakc

(12) am3x¶jqmhcijsxc*g?jJsawj4ngmhnoq_*`hxo

(13) adcqhal »ÆÅ¨ÇhÈ ¢ Á]Á ¯ ¯ Ã Å(ÇÉxmr ¢ Á;Á Ã Ê Çh ª ª j4c?xy¶o?`hxo

(14) xh{bad couxo?jadcakm¦o?`bjp6jJgil}jqo

(15) Gxc*g?jJsawjbxmha|l»¼Ë(ÇbÈ ¢ Á]Á ¯ ¯ Ã Ë(ÇÉxmr ¢ Á;Á Ã Ê Çh ª ª j *c?xyo?`hxoxh{bad¶cto?xo?j*akcakmoi`bj

(16) bjJmhcijsxscPgijqakwjs `bjqgij

(17) Ì·akc@o?`bj-ciprja´r*jJmGo?ginspGysÍ²o[adc jJvkv|¹²Xmbn m oi`rxoam"o?`bjl}ng?wjqg

(18) Jxsctjsso?`bjJg?j4jÎ`bakebao?cp6nscia|o?a sj4mbnmbvkakmbjqxgiaoyxmr¦jJÎXadcto?cvdxc?ciakqxv]jJmGoig?npXyG¹ºcixoiadcl}yXakmb mbnsmbvakmbjxg x jcJ `bavkj[akm4oi`bjvkxotoijqg#qxcijqoi`bjqgij[jJÎ`baeha|ouc0mbjqsxo?a sjmbnmhvakmbjqxgiaoyxmh4jJÎakcto?cmbnmhJvkxsciciadJxv jJmGo?ginspGyG¹² Xansvkxoiakmbfmhnmbvkamhjqxg x jcct{ru` xscjÎphxmhciansm"ci`bnuxmh nwpbg?jqc?ciansmguxg?jlÄxsoiaknm¸@as{bgij § ci`bn coi`bj·guxpb`¨nl4o?`bj3adctnoi`bjqgiw¶xv-vakmbjqcqcixoi{bguxo?ansmÏ{bg? js*xmh=»Ð¡&Çhl}ngo?`bj3 xmÑbjJgzxxvkc jqG{hxoiaknm"nlPcouxoijakm"oi`bj ¥ Æpbvdxmbjs _nctakw~{bvdxo?jpbg?nebvkjJw¶cxg?akciamhl}ginsw!jqmhcijsxsc*hn cqrxcx´hgucoctoijqp0 j~bjJ XakcijqEx xgiadxmGo-nlPoi`bj ÒnXjfÒ

(19) ajqwxmbm·cinvk jqgf Ó²;xmhEakmhJng?prnsg?xoijqEoi`hxoakmso?n" x"Ôn{bmbn G¹oyGp6j~w jJoi`bn3erxcijqEnsmEoi`bj x j¹ pbg?nphxsxoiaknm4l}ng?w~{hvkxoiaknm Õ²P·j¿ oi`bjqmprjqgtl}nsgiwjqfxcijJg?akjqc#nlrnsmbj¹0xmho n¹ºakwjqmhciansmhxvsoijcopbg?nebvkjJw¶c nsmhciakjqgijeGy¶Ö-gisgin Ï×Ø@ÙÚÛ § sÜbsÝhXÞ;ctn~xscPoinf xvkakbxoijo?`bj-phginsprnGctjfwjJoi`bn;[ß-{bwjJg?akqxvhg?jqci{bvo?c þ6ÿ ðà?ò?áäNâãñ±èçã*ú4òuäåNãèuäNåérææ/ëJãäúJåò?äÏãäçã èèuåNézä[ôéèåò?@äêPâå²ãtëòuæ

(20) úì·ä@ò?çäNúJñ±åðîèuöæíÄã÷îãätê#åiïïhò?ê#äê@ñ±èðëúæì òuãó3åñ±?îtábû òuòuórñ ù0ô4?òuãºû}ú äôâûèJõúö[?ñ üûékèãååöñì]ä èæçábåNòuãtñ±öNçöñ±ã÷ñ ó±ãábô4òuðñ ù0ó äNòuñ±çú âòuöãøbó±ètù#0öãùçðñ äN÷âó±ñ±îLê#èèuúqé ûÄì]ì]èúJâüñ±úãºýò

(21) ! "#%$&('*)+!-,/. +#-,$0#-,

(22) +12(3. . . . §.

(23) . .

(24) ¿. T>1 1.2 T=1 G>0 1.1 G=0 C. 5. 4/6. G<0. 1. 4. Vapor Region. Liquid Region 0.9. 0.8 Liquid−Vapor Coexistence Region. 0.7 Saturation Curve 0.6 0.5. 1. 1.5. 798:7<;. 2. 2.5. 3. @¸ as{bg?j § ½4Ôg?xpb`hcnl[o?`bjakcino?`bjJg?w¶xv@vakmbjqcq0c?xoi{hg?xoiaknmÀJ{bgi sjqcq;xmh3» ¡ Çb6l}nsg4x¦ xm·bjJg4xsxvdc-Gxc akm Æpbvkxmbj[³-jJg?j Ladcoi`bj4o?jJwp6jJguxoi{hgijsbxmh xmhE xg?jo?`bjpbg?jqc?ct{hgij4xmh"ciprja´r4 nsv{bwj4xo oi`hj ¥ g?a|o?akxv0couxoijcJg?jqcip6jqo?a sjJvky ¿ ¿ nlctnswj"nloi`bj g?{bmhcxg?j¦phgijctjqmsoakm¨¸@akscq § Ö-clÄxg¶xcfoi`bj svnsehxv*co?gi{roi{hgij"akc¶JnmhJjJg?mbjq0 j nerctjqgi sj

(25) nXnxsgijqjJwjJmGonl0o?`bjcinvk{o?ansmxcnswphxg?jqo?n~o?`bnscij4xpbp6jqxgij¶am¦oi`bjvka|o?jJguxoi{hgijs¿ Pß-nXj-o?`hxo oinvakwa|ooi`hjciaµqj~nl[oi`bjfgijqprnsgto jn¦mbno g?a|o?jbn m3xvkv#o?`bj~akmbaoiadxvnsmhaoiaknmhcÄl § xmhEoi`bj no?`bjJgg?jJvdxo?jqphxguxwjJoijJgucl}ng*o?`bjgi{hmhcJ _n·jJmhoi`badcmbno?j j xmGo~oincixy·oi`rxo~o?`bj ng?pbgijctjqmGo~`bjJg?j¦adc~lÄxgl}g?nw Jnwpbvkjoijs·¸hngfoi`bj ci`bnus¹²guxwpakmGoijqg?xsoiaknm" pbg? nebvkjJw l}nsg

(26) jÎbxwphvjsboi`h jgijctnsv{boiaknm"mbjqxgoi`bj Àxu`Eco?jJw!akcoinXnvkn oin¦erj xehvj4o?n¦gux xmXy"nmrg?joijJnmhJv{rctaknmhcq¸hngo?`hxoho?`bjo?jqu`bmbadG{bjnl@vknJxv@xbxpoiak j4wjqci`Eg?j´hmbjqwmbjo ci`bn{bvd¶e6jakmhJv{hbjqakmo?`bj4xvkng?aoi`bw Eng?jJn sjJgw~{hv|o?akJnwprnsmbjJmGo jJguciansm¶nl0o?`bjxvsng?a|o?`bwBÄl0 § Ç ci`bn{bvderj4l}{bvkvkyoijco?jql}ngwf{bvoiadnwp6nmbjqmGo-jJmhcijsxschn cJ >=. @?. ?. A. CB. DB. ¿ ¿ GÖ

(27) g?g?n nwpb{ouxoiaknmrxvhxmhxvkyctadc@nl;jJmhcijsxscct`hnXuoi{be6jhn uXÞb½ Ý § ÝsÕh Ó§ ÓÞh P g?n m3xmh ]Ö

(28) g?g?n -_ n¹²bawjJmrctaknmhxv#ci`bnu¦o?{be6j~hn l}nsgbjJmhcij~Gxcijqcq krÜÝsÓb½ Ós §s§ § ÓÓX°X P g?n m xmh¿ PÖ

(29) g?g?n Àß

(30) nsmhvdxc?ctadJxvjqmhcij¦sxschn cl}ngctakwpbvkjsjJnwjJoig?y bÜsÞb½ § ÕÝ § ÕÝX°X § ÓÓsÕb P gin mExmr ]Ö

(31) g?g?n Ö-pbpbvkakqxo?ansmEnl jo?`bj¹ 6jJvdn Xadu`¹²_*`hn¿ wphci¿ nmh{badbc

(32) akm3ng?sxmbad Ò

(33) xmbXamhj4JyvkjjqmbakmbjcJ § Þh½ §s§§ Õ §s§ Ýr ÇsÇÇb. EGFIHJFLKMFLNOPFIQ. § ¿. SR JB. Ü Ý. SR fe @R. <T. pqc[rW-h. SR De sR. gXYblXxyf_bk. SR <e @R. VUXWZY[\^]`_babc d. A. ^R fB. SgXhji kmlXno. tR DB. vuLwuLu. A. A. `R fB. R. z. PgXYblXxyf_bkjY{}|~xY@lXkdnYby_byfo|PYJ c x. A.

(34) Ü 1.8. 0.6 Compression Rarefaction. 1.6 0.5. 1.4. Compression Rarefaction. 0.4 u. ρ. 1.2 1 0.8. 0.3 0.2. 0.6. 0.1. 0.4 0.2. 0.4. 0.6. 0. 0.8. 4. 2.5. 3. 2. 2. p. ¿. 0.4. 0.2. 0.4. 0.6. 0.8. 0.6. 0.8. G. 3. 0.2. 1.5. 1. 1. 0. 0.2. 0.4. 0.6. −1. 0.8. x. x. ¸@as{bg?j ½ mhj¹ºawjqmhctaknmrxv;jJmrctj¦Gxc*nswph{o?xoiaknm0½Îbxwphvj vnonloi`hj4bjJmhcia|oy r sjJvknXJa|oy # pbg?jqc?ct{hgij hxmh¦oi`bj4l}{bmrbxwjJmGouxvjqgiak xo?a sjnl@Gxc*yXmhxwadJc

(35) » § . e. 1.2. . . Compression Rarefaction. G=0 1.1. . . 1. Contact Discontinuity. . S = 2.7439 0.9 Saturation Curve. S = 0.9802. 0.8 Liquid−Vapor Coexistence Region 0.7. Â ¸@as{bg?jÜh½ mhj¹ºawjqmhctaknmrxv;jJmrctjsxsc*nwpb{bo?xo?ansm0½Îbxwpbvkj § [vkno*nl#oi`bjcinvk{oiaknmamoi`bj ¥ pbvdxmbjs[³

(36) jqgijÌ(jJmbnoijcoi`bjciprja´r4jJmGo?ginspGys 0.6 0.5. . 1. 1.5. ?. 2. 2.5. Me. 3.

(37) Ý Compression Rarefaction. 0.85. 0.12. 0.8. 0.1. 0.75. 0.08 u. ρ. Expansion Shock. 0.7. 0.06. 0.65. 0.04. 0.6. 0.02 0.2. 0.4. 0.6. 0. 0.8. 0.2. 0.4. 0.6. 0.8. 0.6. 0.8. −0.2. 1.05. −0.4 p. G. 1. −0.6 −0.8. 0.95 −1 −1.2. 0.9 0.2. 0.4. 0.6. ¿. 0.8. 0.2. x. 0.4 x. ¸@as{bg?jÝr½ mhj¹ºawjqmhctaknmrxv;jJmrctj¦Gxc*nswph{o?xoiaknm0½Îbxwphvj vnonloi`hj4bjJmhcia|oy r sjJvknXJa|oy # pbg?jqc?ct{hgij hxmh¦oi`bj4l}{bmrbxwjJmGouxvjqgiak xo?a sjnl@Gxc*yXmhxwadJc

(38) » . e. . . 1.2 G=0 1.1. 5. 4/6. Compression Rarefaction. 1 Expansion Shock. 4. 0.9 S = 2.343. Saturation Curve 0.8. S = 0.43. Liquid−Vapor Coexistence Region 0.7. 0.6 0.5. 1. 1.5. 798:7<;. 2. ¿. 2.5. 3. @¸ as{bg?j4b½ mhj¹ºawjqmhctaknmrxv;jJmrctjsxsc*nwpb{bo?xo?ansm0½Îbxwpbvkj [vkno*nl#oi`bjcinvk{oiaknmamoi`bj ¥ pbvdxmbjs[³

(39) jqgijÌ(jJmbnoijcoi`bjciprja´r4jJmGo?ginspGys . Me.

(40) 0.35 0.8 0.7. 0.25 0.2. 0.6. u. ρ. 0.3. Compression Rarefaction. 0.15. 0.5. 0.1 0.4 0.05 0.3 0.2. 0.4. 0.6. 0. 0.8. 0.2. 0.4. 0.2. 0.4. 0.6. 0.8. 0.6. 0.8. 0.6 1 0.4 p. G. 0.9 0.2. 0.8 0 0.7 −0.2 0.6 0.2. 0.4. 0.6. 0.8. x. x. @¸ as{bg?j~Þb½ mbjJ¹²akwjJmhciaknmhxvjJmrctjfsxc-nwpb{bo?xo?ansm0½Îbxwpbvkj~Üh [vkno-nloi`bjjJmrctaoy 6] jqvnaoy # pbg?jqc?ct{hgij hxmh¦oi`bj4l}{bmrbxwjJmGouxvjqgiak xo?a sjnl@Gxc*yXmhxwadJc

(41) » L. Le. . . 1.2. Compression Rarefaction 1.1 G=0. . . 1. . 0.9 Saturation Curve. S = 2.343. 0.8 S = 0.989 Liquid−Vapor Coexistence Region 0.7. Â ¸@as{bg?j°½ mhj¹ºawjqmhctaknmrxv;jJmrctjsxsc*nwpb{bo?xo?ansm0½Îbxwpbvkj4Üb [vkno*nl#oi`bjcinvk{oiaknmamoi`bj ¥ pbvdxmbjs[³

(42) jqgijÌ(jJmbnoijcoi`bjciprja´r4jJmGo?ginspGys 0.6 0.5. . 1. 1.5. ?. 2. 2.5. Me. 3.

(43) Þ. ¸@as{bg?j¦Õb½_ n¹ºakwjqmhciansmhxvbjJmhcij¦Gxc~Jnwpb{ouxo?ansm0½ X`bnuG¹guxwpg?jhjo?ansm jÎbxwpbvkj nmGo?n{bg pbvknoucnl;oi`bj jJmhciaoyxmh¶pbg?jqc?ci{bgijssJginGcict¹²cijqoiaknmpbvknonl;oi`hjcinvk{oiaknm¦xvnsmbo?`bj-guxwp0Xxmh§ pbvnonl;oi`bj cinvk{oiaknm"am"o?`bj ¥ ¼phvkxmbjxg?j4ci`bn m0 . T. 0.2. 0.2. 0.1. 0.1. 0. 0. 0.2. 0.4. 0.6. 0.8. 0. 1. 1. 0. 0.2. 0.4. 0.6. 0.8. 1. 1.1 ρ G. 1 p. 0.5. 0.9. 0 0.8 −0.5. 0.7 0.6. ¿ @¸ as{bg?j¦Ób½_ n¹ºakwjqmhciansmhxvbjJmhcij¦Gxc~Jnwpb{ouxo?ansm0½ X`bnuG¹guxwpg?jhjo?ansm jÎbxwpbvkj nmGo?n{bg pbvknoucnl;oi`bjjJmhciaoyxmh¶pbg?jqc?ci{bgijssJginGcict¹²cijqoiaknmpbvknonl;oi`hjcinvk{oiaknm¦xvnsmbo?`bj-guxwp0Xxmhpbvnonl;oi`bj cinvk{oiaknm"am"o?`bj ¥ ¼phvkxmbjxg?j4ci`bn m0 0. 0.2. 0.4. 0.6. 0.8. 1. 1. 2. 3. 4. V. . T.

(44) °. ¸@as{bg?j Çb½_ n¹²bawjJmrctaknmhxv@bjJmhcijsxc4Jnwpb{ouxoiaknm½ `bnus¹²guxwpÀg?jhjo?ansmÀjÎbxwphvj¶Üb nmGo?n{bg pbvknoucnl;§ oi`bjjJmhciaoyxmh¶pbg?jqc?ci{bgijssJginGcict¹²cijqoiaknmpbvknonl;oi`hjcinvk{oiaknm¦xvnsmbo?`bj-guxwp0Xxmhpbvnonl;oi`bj cinvk{oiaknm"am"o?`bj ¥ ¼phvkxmbjxg?j4ci`bn m0 . T.

(45) Õ. ¸@as{bg?j ½_ n¹²bawjJmrctaknmhxv@bjJmhcijsxc4Jnwpb{ouxoiaknm½ `bnus¹²guxwpÀg?jhjo?ansmÀjÎbxwphvjÝh nmGo?n{bg pbvknoucnl;§soi§ `bjjJmhciaoyxmh¶pbg?jqc?ci{bgijssJginGcict¹²cijqoiaknmpbvknonl;oi`hjcinvk{oiaknm¦xvnsmbo?`bj-guxwp0Xxmhpbvnonl;oi`bj cinvk{oiaknm"am"o?`bj ¥ ¼phvkxmbjxg?j4ci`bn m0 . T.

(46) Ó. ¿ @¸ as{bg?j § ½_ n¹²bawjJmrctaknmhxv@bjJmhcijsxc4Jnwpb{ouxoiaknm½ `bnus¹²guxwpÀg?jhjo?ansmÀjÎbxwphvj nmGo?n{bg pbvknoucnl;oi`bjjJmhciaoyxmh¶pbg?jqc?ci{bgijssJginGcict¹²cijqoiaknmpbvknonl;oi`hjcinvk{oiaknm¦xvnsmbo?`bj-guxwp0Xxmhpbvnonl;oi`bj cinvk{oiaknm"am"o?`bj ¥ ¼phvkxmbjxg?j4ci`bn m0 . T.

(47) §Ç ~Ò n{hg?xmsoxmh fX¸bg?ajgiadu`hcq Ïakvjqys¹²ÍºmGoijqg?c?akjJmhJjXß

(48) j [nsgi] § ÓÝsÕb Þ ]³~¸bjqgiGxcinm0_4 [³-nh Ö-gisgin xmrfÔ[w¶xmG{hjJvb_*`bjJnsgiyl}ngP¿ pbg?n{hJamb4x4ctakmbsvjJ¹pb`rxcij guxg?jlÄxsoiaknmct`bnu x jakm xci`bnu¶oi{herjs khÝÝX½ ÜG° Ýr ÇÇ § ±°~-b¸bjJg?waN n jJgß

(49) j [nsgi] § ÓGÞh Õ~Ò h 0j jG{bj"x sj¿ pbginsphxGxo?ansmxvsng?a|o?`bw¶cl}ngwf{bv|o?a|¹ºakwjJmhciansmhxv`XyGp6jJg?e6nvkak¶ciyco?jJw¶cJ k § Ü § ½ Ü ° ÜGÜb § ÓÓX°X Ó P [qÒ

(50) nGjssÖ-pbpbg?nÎX¿ akw¶xo?j[Òajqw¶xmbm~cinvk jqg?cqJprxguxwjo?jJg0 sjqo?ngxmha ]jJg?jJmhJjciu`hjJwj khÝsÜh½ ÜGs° ÜG° § ÓÕ § § Ç ¹ X`XyX{bjs#Ö/h{badX¹waÎXoi{bg?j-oyXp6jxvsng?a|o?`bw¼l}ngnswpbgijciciakebvj-w~{bvoiadnswp6nmhjJmGo*hn Ï a|o?`¦ xm jqgÑxxvkc*js{rxoiaknm"nlPcouxo?j k § Þh½ ÝGÜ ÕsÕb § ÓÓsÓb ZT. Z. DU<l-c xd%YJy-n[Li kYJ_Jy@oUXW-Y2[\ _Ja*c d. . . b. :R JB. LgXhfi kmlXnopqc[rW-h. fW-c xY2oJby@_Jn[ d 9. A. . j s %. vgXh. YbL@lX h¡| WX*d h. A. Se . ¢. | WX*d%h. C. £B. A. ¤. IgXh YJIflX£hj| W¤*d%h. A. -gXh YJIflX£h.

(51)