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Navier-Stokes Equation

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Navier-Stokes Equation ,90HWHRURORJLFDO'\QDPLFV ,9 ,QWURGXFWLRQ ,9)RUFHV DQG HTXDWLRQ RI PRWLRQV ,9$WPRVSKHULFFLUFXODWLRQ IV/1 ,90HWHRURORJLFDO'\QDPLFV ,9 ,QWURGXFWLRQ ,9)RUFHV DQG HTXDWLRQ RI PRWLRQV ,9$WPRVSKHULFFLUFXODWLRQ IV/2 Dynamics: Introduction ,9,QWURGXFWLRQ y GHILQLWLRQ RI G\QDPLFDOPHWHRURORJ\ ÎUHVHDUFK RQ WKH QDWXUHDQGFDXVHRI DWPRVSKHULFPRWLRQV y WZRILHOGV ÎNLQHPDWLFV Ö VWXG\ RQQDWXUHDQG SKHQRPHQD RIDLU PRWLRQ ÎG\QDPLFV Ö VWXG\ RI FDXVHV RIDLU PRWLRQV :HZLOOPDLQO\FRQFHQWUDWH RQ WKH VHFRQG SDUW G\QDPLFV IV/3 Pressure gradient force ,9)RUFHV DQG HTXDWLRQ RI PRWLRQ K KKdv y 1HZWRQµVODZ FFm==⋅∑ i i dt y )ROORZLQJDWPRVSKHULFIRUFHVDUHLPSRUWDQW ÎSUHVVXUHJUDGLHQWIRUFH 3*) ÎJUDYLW\ IRUFH ÎIULFWLRQ Î&RULROLV IRUFH IV/4 Pressure gradient force ,93UHVVXUHJUDGLHQWIRUFH y 3UHVVXUH IRUFHDUHD y )RUFHIURPOHIW =⋅ ⋅ Fpdydzleft ∂p F=− p + dx dy ⋅ dz right ∂x ∂∂pp y VXP RI IRUFHV FFF= + =−⋅⋅⋅=−⋅ dxdydzdV pleftrightx ∂∂xx ∂∂ y )RUFHSHUXQLWPDVV −⋅pdV =−⋅1 p ∂∂ρ xdmm x ρ = m m V K 11K y *HQHUDO f=− ∇ p =− ⋅ grad p p ρ ρ mm 1RWHXQLWLV 1NJ IV/5 Pressure gradient force ,93UHVVXUHJUDGLHQWIRUFH FRQWLQXHG K 11K f=− ∇ p =− ⋅ grad p p ρ ρ mm K K ∇p y SUHVVXUHJUDGLHQWIRUFHDFWVÄGRZQKLOO³RI WKHSUHVVXUHJUDGLHQW y ZLQG IRUPHGIURPSUHVVXUHJUDGLHQWIRUFHLVFDOOHG(XOHULDQ ZLQG y WKLV W\SH RI ZLQGVDUHIRXQG ÎDW WKHHTXDWRU QR &RULROLVIRUFH ÎVPDOOVFDOH WKHUPDO FLUFXODWLRQ NP IV/6 Thermal circulation ,93UHVVXUHJUDGLHQWIRUFH FRQWLQXHG y7KHUPDOFLUFXODWLRQLVFDXVHGE\DKRUL]RQWDOWHPSHUDWXUHJUDGLHQW Î([DPSOHV RYHQ ZDUP DQG ZLQGRZ FROG RSHQILHOG ZDUP DQG IRUUHVW FROG FROGODNH DQGZDUPVKRUH XUEDQUHJLRQ ZDUP DQG JUHHQVXUURXQGLQJV FROG FRRORFHDQ DQGZDUPODQG VHHEUHH]H ZDUPRFHDQ DQG FROG ODQG QLJKWZLQWHU yLQVRPHFDVHV WKHUPDO FLUFXODWLRQDOVRSOD\VDQLPSRUWDQWUROH DW ODUJHUVFDOHV ÎWUDGHZLQGV HTXDWRU DQGFRROVXEWURSLFV ÎPRQVRRQ ,QGLDQ2FHDQDQG,QGLDQVXEFRQWLQHQW EXWQRWHWKDW&RULROLVIRUFHDOVRFRQWULEXWHWRWKHVHW\SHV RI FLUFXODWLRQ IV/7 Thermal circulation ,93UHVVXUHJUDGLHQWIRUFH FRQWLQXHG %DURFOLQLF FRQGLWLRQ LQGXFH FLUFXODWLRQ :DUQHFNH IV/8 baroclinity and thermal circulation : : 3UHVVXUHJUDGLHQWIRUFH EDURFOLQLW\ :DUQHFNH IV/9 Gravity force ,9*UDYLW\ IRUFH K K y 1HZWRQ ODZ =⋅ Fmgg KKK y JUDYLW\IRUFH =− =− ⋅ fggkg 1RWHXQLWLV 1NJ y VXUIDFHV RI HTXDOJHRSRWHQWLDOKHLJKW ] H[SHULHQFHWKHVDPHJUDYLWDWLRQDO IRUFH 1(,)h ghϕ ÎJHRSRWHQWLDOKHLJKW zdhghh=≈⋅∫ '(,')ϕ h´0= ggoo Î*HRSRWHQWLDO φ =⋅=ϕ ⋅ gzo g(,) hh 0 KKKKKK =−∇ ⋅φ =−∇ ⋅φ φ = f gz h 0 ⋅ gzo IV/10 Friction ,9)ULFWLRQ y )ULFWLRQIRUFHVDSSHDUZKHQWKH ZLQG ILHOGLVVKHDUHG YHUWLFDOVKHDULQJHJMHWVWUHDP KRUL]RQWDOVKHDULQJWRSRJUDSK\VXUIDFHURXJKQHVV y 6ORZLQJGRZQRIDLU SDUFHOGXHWRWXUEXOHQFHVSHUSHQGLFXODUWRWKHPDLQ ZLQG GLUHFWLRQ y 6ORZ DLU SDUFHOGHFHOHUDWHIDVWHUPRYLQJ DLU SDUFHOV FROOLVLRQV DQGVRRQ LQWURGXFLQJDVKHDULQJ LQWKH ZLQG ILHOG IV/11 Coriolis force ,9&RULROLVIRUFH y 1HZWRQµVODZLVRQO\YDOLGLQDV\VWHP ZKLFKLV DW UHVWRULVLQXQLIRUPPRWLRQ FRQVWDQWYHORFLW\HJFRQVWDQW VSHHG DQG FRQVWDQW GLUHFWLRQ y (DUWK LVURWDWLQJDURXQGLWVD[LV DQGWKHUHIRUHLVDQDFFHODUDWLQJ V\VWHP Î:HKDYHWRWDNHLQWRDFFRXQWHDUWKµVURWDWLRQ LI 1HZWRQVODZ RI PRWLRQ LVDSSOLHG dv y ([DPSOH5RXQGDERXW mmrmvr=⋅ω 22 ⋅=⋅/ dt Î([WHUQDOYLHZHUVHHVDIRUFHDFWLQK JWRZDUGVWKHFHQWHU &HQWULSHGDOIRUFH Fc ω = 2/π T Î3HUVRQRQ WKHURXQGDERXWIHHOVWKHPRPHQWK RI LQHUWLD &HQWULIXJDOIRUFH * Fc KK Î&RULROLVIRUFHLVDYLUWXDOIRUFH +=* FFcc0 IV/12 Coriolis force ,9&RULROLVIRUFH FRQWLQXHG y ([DPSOH 6RXWKQRUWKPRWLRQ IV/13 Coriolis force y ([DPSOH 6RXWKQRUWKPRWLRQ FRQWLQXHG ()ϕϕ+∆ − ϕ ϕ avfv=Ω⋅2sinϕ ⋅= ⋅ cos cos → dcos ∆ϕϕd f ≡Ω⋅2 sinϕ Coriolis parameter IV/14 Coriolis force ,9&RULROLVIRUFH FRQWLQXHG y *HQHUDOGHVFULSWLRQ RI PRWLRQ DFFHOHUDWLRQ LQDURWDWLQJHDUWK V\VWHP K K dv dv KKKKK =a −2 ⋅() Ω×vr −Ω× () Ω× dt dt &RULROLVIRUFH DFFHOHUDWLRQREVHUYHG &HQWULIXJDO IRUFH IURP DIL[HGORFDWLRQ RQ HDUWK DFFHOHUDWLRQLQWKHLQHUWLDO V\VWHP GXH WRUHDOIRUFHVHJ3*)JUDYLW\ K K K =− ⋅ Ω× y &RULROLV IRUFH fvC 2 () KKK y &HQWULIXJDOIRUFHWHUPLVFRQWDLQHGLQWKHFRUUHFWLRQIRUΩ×() Ω×rr =Ω2 JUDYLW\DFFHOHUDWLRQ DQG LVQRZSDUW RI WKHPRGLILHGJUDYLW\ IRUFH VHH JHSRWHQWLDOKHLJKWGHILQLWLRQ IV/15 Coriolis force ,9&RULROLVIRUFH FRQWLQXHG K K K =− ⋅ Ω× y /HWµVORRN LQWRWKHYDULRXVWHUPVFRQWDLQHGLQWKH&RULROLVIRUFH fvC 2 () u K vv= &DOFXODWHGOLNHDGHWHUPLQDQW RIDPDWUL[ w KK iuΩ i0 u KKK K K Kx K K K ffvgkjvgkj+ =Ω×−⋅=−22cos Ω −⋅=− Ω⋅ϕ vgk −⋅= CgKK y ΩΩ⋅ϕ kwz ksin w K =Ω2sincos()vwi ⋅ϕ − ⋅ϕ ⋅ K wuv , −Ω ⋅ϕ ⋅ + 2sin()uj 2QO\KRUL]RQWDO w ≈ 0 KK FRPSRQHQW RI &RULROLV 2Ω⋅ug +Ω2cos()ukgk ⋅ϕ ⋅ − ⋅ = IRUFHLPSRUWDQW 2sinΩ⋅v ⋅ ϕ KKK ≈Ω2()viujgku ⋅ sinϕ ⋅−Ω 2 () ⋅ sinϕ ⋅− ⋅ =−Ω⋅⋅ 2 sinϕ IV/16 −g Coriolis force ,9&RULROLVIRUFH FRQWLQXHG y &RULROLV IRUFHFDQEHQRZVLPSOLILHGDVIROORZV KKK K 2sinΩ⋅ϕ vv fvf=−2 ⋅() Ω× ≈() = = f = CCH −Ω2sinϕ ⋅uu − KK KK =⋅−⋅=−× fv i fu j f() k vH 2QO\KRUL]RQWDOYHORFLW\FRPSRQHQWLPSRUWDQW &RULROLVSDUDPHWHU K u v = f ≡Ω⋅2sinϕ H v K k K KKK cab=× vH K K × K K kvH −× kvH y &RULROLV IRUFHGLUHFWVWKH ZLQGWRWKH ULJKW IURPLWV RULJLQDOGLUHFWLRQJLYHQE\ Y+ LQWKH1+ WRWKHOHIW LQ6+ IV/17 Navier-Stokes Equation ,91DYLHU6WRNHV(TXDWLRQ KK dv∂ v KKKKKKKK1 K =+∇==−∇−⋅−Ω×+()vvf pgk2() v f dt∂ t ρ F m ,,,,,,,9 Î, SUHVVXUHJUDGLHQW IRUFH Î,, JUDYLW\ IRUFH LQFOXGLQJFRUUHFWLRQIRUFHQWULIXJDO IRUFH Î,,, &RULROLV IRUFH Î,9 IULFWLRQ IRUFH K K ∂ KKK y $FFHOHUDWLRQ RIDLU SDUFHOORFDOGHULYDWLYHDQGdv v DGYHFWLRQWHUP ()vv∇ dt ∂t y ,IZHFRQVLGHU DLU DVDSHUIHFWOLTXLG K\GURG\QDPLFV WKHQIULFWLRQIRUFHLV ]HURl (XOHULDQHTXDWLRQ RI PRWLRQ y 1DYLHU6WRNHVLVWKHEDVLFHTXDWLRQ RI WKHG\QDPLFDOPHWHRURORJ\ KKK y %HFDXVH RIWKHHTXDWLRQ()vv∇ RI PRWLRQLVDGLIIHUHQWLDOHTXDWLRQ RIVHFRQG RUGHUDQG FDQQRWEHVROYHGDQDO\WLFDOO\l QXPHULFDOVROXWLRQQHHGHG IV/18 Navier-Stokes Equation ,91DYLHU6WRNHV(TXDWLRQ FRQWLQXHG KK dv∂ v KKKKKKKK1 K =+∇==−∇−⋅−Ω×+()vvf pgk2() v f ∂ ρ F dt t m y LQFDUWHVLDQFRRUGLQDWHV GX∂∂∂ X X X ∂ X ∂ S =+XYZ + + =−D DDD +ΩVLQFRV() Yϕϕ − Z + I)[ GW∂∂∂ W [ \ ∂ ]ρP ∂ [ GY∂∂∂ Y Y Y ∂ Y ∂ S =+XYZ + + =−D DDD −Ω⋅VLQ Xϕ DDDDDDDD + I)\ GW∂∂∂ W [ \ ∂ ]ρP ∂ \ GZ∂∂∂ Z Z Z ∂ Z ∂ S =+X +YZ + =−−−Ω⋅ JFRV XϕDDDDDDDD +I)] GW∂∂∂ W [ \ ∂ ]ρP ∂ ] ,,,,,, ,9 y $OOEDVLFHTXDWLRQV RI K\GURG\QDPLFVWKHUPRG\QDPLFVWKDWDUHLPSRUWDQWIRU DWPRVSKHULFG\QDPLFVDUH IV/19 Navier-Stokes Equation ,91DYLHU6WRNHV(TXDWLRQ FRQWLQXHG y %DVLFHTXDWLRQV RI K\GURG\QDPLFVWKHUPRG\QDPLFVWKDWDUHLPSRUWDQWIRU DWPRVSKHULFG\QDPLFVDUH ÎHTXDWLRQ RI PRWLRQ 1DYLHU6WRNHV LQWKUHHFRRUGLQDWHV[\] ÎFRQWLQXLW\HTXDWLRQ l PDVVFRQVHUYDWLRQ ∂ρ K K m =−∇()ρ ⋅v ∂t m ÎZDWHUYDSRUEXGJHW ÎVWODZ RIWKHUPRG\QDPLFV ÎJDVODZV LGHDOJDV3RLVVRQHTXDWLRQUHODWHGWRDGLDEDWLFSURFHVVHV y 7KHVHEDVLFODZVDUHUHTXLUHGLQQXPHULFDOPRGHOV LQDGGLWLRQWKHUHPD\EHD QHHGIRUFKHPLFDOWHUPVEXWQRWQHHGHGLQZHDWKHUSUHGLFWLRQPRGHOV IV/20 Scale analysis ,96FDOHDQDO\VLV y 6LQFHHTXDWLRQ RI PRWLRQDUHGLIILFXOWWRVROYHRQHFDQWU\WRVLPSOLI\HTXDWLRQV E\DQDO\VLV RI VFDOHV PDJQLWXGH RIRUGHURI YDULRXVWHUPV y +HUHZHFRQVLGHUPLGODWLWXGHV\QRSWLFVFDOHV/ NP&KDUDFWHULVWLF YDOXHVDUH ÎKRUL]RQWDOZLQG VSHHG8 9 PV Î9HUWLFDO ZLQG VSHHG: PV Î+HLJKW WURSRVSKHUH + NP Î3UHVVXUHYDULDELOLW\%3 K3D Î7LPH VFDOH7 /8a V K Î&RULROLVSDUDPHWHUI 8VLQK a QV VLQ a; V ÎSP NJP Î* PV Î3 K3D IV/21 Scale analysis ,96FDOHDQDO\VLV IV/22 Geostrophic approximation ,9$SSUR[LPDWLRQV y ,QPRVWDSSUR[LPDWLRQVDEDODQFH RI IRUFHVLVDVVXPHG HTXLOLEULXP VWDWHQR DFFHOHUDWLRQ HJ KK == ff∑ i 0 ,9JHRVWURSKLFDSSUR[LPDWLRQi KK += ()ffCp () 0 HHKKKK K K ⇒ −ρ −−11 ∇− Ω× =−ρ ∇− × = mHmHpv2( ) pfkv ( ) 0 K K ()fv⊥ CHH K K ()fv⊥ pHH K 1 K K y *HRVWURSKLF ZLQG FRPSRQHQW vkp=×∇⋅() gsρ ⋅ H m f IV/23 Geostrophic approximation K 1 K K ,9JHRVWURSKLFDSSUR[LPDWLRQ FRQWLQXHG vkp=×∇⋅() gsρ ⋅ H m f y *HRVWURSKLFDSSUR[LPDWLRQLVVXIILFLHQWO\YDOLGDERYHSODQHWDU\ ERXQGDU\OD\HU DERYHNPDOWLWXGH y $W ORZODWLWXGHV QHDUHTXDWRU JHRVWURSKLF ZLQG VSHHG YJV LQFUHDVHVVLQFH WKH&RULROLVSDUDPHWHU I DSSURDFKHVEXWSUHVVXUHJDUGLHQWIRUFHDOVR WHQGVWR]HUR,QJHQHUDOJHRVWURSKLFZLQGVDUHQRW ZHOOGHILQHGLQWKHWURSLFV KKK ++ = I FHQWULIXJDOIRUFH fffCpZ0 ] IV/24 Cyclostrophic wind ,9F\FORVWURSKLF ZLQG y QR &RULROLVIRUFH HTXDWRU y QRIULFWLRQ IRUFH RFHDQ y 2QO\SUHVVXUHJUDGLHQWIRUFH IS DQG FHQWULIXJDOIRUFH I= EDODQFH KK ()ff+= () 0 pZHH 1 ∂pv2 −+=0 ρ ∂ m nr y $QWLF\FORQLF DQG F\FORQLFPRWLRQVDUHDOORZHG y ([DPSOHWURSLFDOVWRUPV KXUULFDQHVW\SKRRQV Î UaNP Î IRUP PDLQO\QHDU ODWLWXGHV Î LQPLGGOHODWLWXGHVWKH\GHYHORSLQWRZHDNHUF\FORQHV PXVWDFFRXQWIRU JHVWURSKLF ZLQG FRPSRQHQW KHUH IV/25 Ageostrophic wind ,9F\FORVWURSKLF ZLQG FRQWLQXHG y ([DPSOHWRUQDGRHV Î UaP ,9DJHRVWURSKLF ZLQG IULFWLRQIRUFH y B GHFUHDVHVZLWKDOWLWXGH y B DW VXUIDFHLVDIXQFWLRQ RI URXJKQHVV OD\HUVWDELOLW\DQG JHRJUDSKLFDOODWLWXGH y *OREDODYHUDJHQHDUVXUIDFHB IV/26 ageostrophic wind ,9DJHRVWURSKLF ZLQG FRQWLQXHG y VXUIDFHIULFWLRQWHQGWRGHFUHDVHWKH SUHVVXUHJDUGLHQWV ILOOORZV DQG HPSW\ KLJKV y /RZVDUHILOOHGIDVWHU RQODQG WKDQ RQ RFHDQ ODQG LVURXJKHUWKDQRFHDQ y /RZVDUHSUHIHUDEO\IRUPHGRYHURFHDQ y 7URSLFDOVWRUPVGHYHORSRQO\RYHURFHDQ DQG TXLFNO\GLVVLSDWHRYHU ODQG y &RQWLQXLW\HTXDWLRQUHTXLUHVWKDWLQ /RZV DLU LVULVLQJ DQG VLQNLQJLQ+LJKV IV/27 ageostrophic wind ,9DJHRVWURSKLF ZLQG FRQWLQXHG y 6XUIDFH/RZ Ö ULVLQJ DLU PDVVÖ SUHFLSLWDWLRQ DQG FRQGHQVDWLRQ y 6XUIDFHKLJKÖ VLQNLQJ DLU PDVVÖ GLVVROYLQJFORXGVVWDELOLVLQJOD\HULQJ LQYHUVLRQ IV/28 Vertical motion ,99HUWLFDOPRWLRQ KK ()ff+= () 0 gpVV 1 ∂p ⇒−g − =0 y OHDGV WRK\GURVWDWLFHTXDWLRQ ρ ∂ m z ∂p ⇒=−⋅ρ g ∂z m IV/29 ,9$WPRVSKHULFFLUFXODWLRQ 2QH FHOOFLUFXODWLRQ E\+DGOH\ 7KUHHFHOOFLUFXODWLRQ E\%HUJHURQ IV/30 ,9$WPRVSKHULFFLUFXODWLRQ IV/31 ,9$WPRVSKHULFFLUFXODWLRQ $KUHQV 7XUFR IV/32.
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