Convection Derivatives February 17, E+01 1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06 1.E-07 1.E-08 1.E-09 1.E-10. Error

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1 onvcton rvtvs brry 7, nt Volm Mtho or onvcton rvtvs Lrry rtto Mchncl ngnrng 69 omttonl l ynmcs brry 7, Otln Rv nmrcl nlyss bscs oncl rslts or son th sorc nlyss Introc nt-volm mtho or convcton Not n or vrg φ vl t cll cs or convcton trms mn s o rthmtc mn s vrg vl n sho ts roblms Rv Nmrcl nlyss nt-rnc rssons orr, bckr, cntrl Trncton rror rom Tylor srs roortonl to st-sz, h, to n th or s cll n th orr rror ε h ) nt rnc grs, y, z k, t n th (, y, z k, t n ) n jk Us Tylor srs to gt rssons or rvtvs on grs Grtr rror on non-norm grs 3 rst rvtv rssons rst orr orr rst orr bckr Scon orr cntrl Scon orr orr Scon orr bckr ' h ' h h ' 4 h h) h) 3 h ) ' 4 h 3 ' O h O h 4 Othr rssons '' Scon rvtv h ) rthmtc mn h h ) Unvn st szs h ; r h /h 3 3 '' h ''' ' h h h...! h h 3! h h ( r ) r '' h h h 3 3 ''' h h '' '' h h... 3 h h h h 5 rror Trncton gr -. ct o n St Sz Rono on rror Rono rror Scon-orr Trncton rror St Sz log ε log 5 3 ε 7 n 5 log 6 log h h M 69 omttonl l ynmcs

2 onvcton rvtvs brry 7, nt-volm roch Intgrt ovr smll volm hr rvtvs occr, rlc thm by nt-rnc rssons hr sorc trms occr, rlc trms lk ΔV SV by S vg ΔV S vg s vrg S or control volm nt Volm gr notton Vrbls n t Nos Volm bonrs t cs / / δ, δ, δ, δ nt Volm ml onsr son n sorc only S S V S Intgrt ovr nt-volm gr / / δ, δ, δ, δ SV S Δ V SΔV ht r n φ/ t n cs? 7 8 nt Volm ml II Us mn vl or cocnt ( )/ n ( )/ Us cntrl rnc or φ/ φ/ (φ φ )/δ φ/ (φ φ )/δ Sbsttt nto S Δ V n SΔV S S S S S S δ δ δ δ δ δ n S S δ δ δ δ S S nt-volm ml III Ths rrsnts qton or on no N to rt qtons or ll nos Sml or on-mnsonl roblm Hr s hr,, - notton s sr, φ -, φ, φ S, ly ths rslt or nos rom to N Nos n N r bonry contons ml: constnt rorty vls: kg/m s m kg m/s δ L/N th L m ( m ) kg m s δ m N N S 9 kg s nt-volm ml IV onsr sorc trm Lt S βφ th β kg/m 3 s Gt vrg sorc trm S V βv β βδ Δ ΔV δ Gt vls or S n S rom nton SΔV S S SΔV βδ S n S βδ rom m, δ ( m)/n, n β kg/m 3 s, S βδ ( kg/s) / N nt-volm ml V Gt nmrcl vls or ll cocnts kg S S S N s kg kg N S N s N s Sbsttt nto qton n rrrng N N N N N N to hnl bonry contons Lt φ lt t n φ rght 3 t L M 69 omttonl l ynmcs

3 onvcton rvtvs brry 7, nt-volm ml VI Nol vls φ φ lt n φ N φ rght 3 st by bonry contons t ( no n rom lt bonry) φ ( /N )φ φ th φ φ lt gvs ( /N )φ φ φ lt t N ( no n rom rght bonry) φ N- ( /N )φ N- φ N th φ N φ rght 3 gvs φ N- ( /N )φ N- φ rght 3 Gnrl qton ls t ll othr nos ck N 5 so /N /5 /5.4 n /N.4 Solv qtons or,, 3, n 4 3 nt-volm ml VI Systm o qtons to solv.4φ φ Mtr orm.4 φ.4φ φ.4 φ3.4φ3 φ3.4 φ4.4φ4 3 φ φ φ3.4 φ Nmrcl n ct Solton.5 Nmrcl rror.5..5 ct Nmrcl bsolt rror.5. N 5 N N rror n Grnt ct Η/ t s Scon orr rst Orr Η/ rror Η/ rror N N N ct Η/ t L s Scon orr rst Orr Η/ rror Η/ rror N N N onvcton Trms Sty qton th convcton n son trms n on mnson Sty contnty qton n on mnson ly nt volm roch to ntgrt smll volm V V V 8 M 69 omttonl l ynmcs 3

4 onvcton rvtvs brry 7, M 69 omttonl l ynmcs 4 9 Rv nt-volm roch Intgrt ovr smll volm hr rvtvs occr, rlc thm by nt-rnc rssons hr sorc trms occr, rlc trms lk ΔV SV by S vg ΔV S vg s vrg S or control volm nt Volm gr notton / / δ, δ, δ, δ Vrbls n t Nos Volm bonrs t cs Intgrton o s V V V Intgrton o s II Look t constnt rs so tht r ros ot o qton Us rvos rslts or son trms δ δ n n /δ Intgrton o s II Ho o rlt c vls to nol vls or convcton trms? rst thoght: s (scon-orr ccrt) rthmtc mn or c vls 3 Intgrton o s III Rvs φ cocnt to ncl sm o φ n φ cocnts s zro 4 ml roblm Solton onstnt,, n th φ φ t n φ φ L t L

5 onvcton rvtvs brry 7, ml roblm Solton II t bonry conton φ φ t n φ φ L t L to gt n L L L L L L Sbsttt n nto φ() ( L) L L 5 ml roblm Solton III L L L L L L L L L L L L L L L/ 6 (ξ - ξ)/(ξ L - ξ) ct Solton ly nt-volm qton ( ) In ths ml n r constnts th constnt n cocnt Gr s norm so δ s constnt δ δ Bonry contons t n L rnt or convcton n son 8 Bonry ontons t (n L) kno φ φ lt (n φ φ rght ) n o not n mn vl qton or φ n convcton trm lt ( ) lt rght ( ) ( ) lt lt δ δ δ lt 9 Bonry ontons II qton or lt bonry lt 3 qton or rght bonry rght rght rght lt lt lt rght rght ( ) 3 M 69 omttonl l ynmcs 5

6 onvcton rvtvs brry 7, Bonry ontons III rght N N rght N 3 ( ) ( ) rght t: L m,. m/s,. kg/m s, δ. m, kg/m 3 ( kg/m 3 )(. m/s). kg/m s /δ (. kg/m s)(. m).5 kg/m s ( ) ( ) rght N N N ( ) rght 3 N qtons Non-bonry qtons th,.5 kg/m s, n. kg/m s /.45kg / m s /.55kg / m s.45kg / m s.55kg / m s kg / m s Lt-bonry qton ( ) lt lt 3 qtons II Rght-bonry qton N 3 N. 55N.45N. 9 ( ) rght St.5 kg/m s,. kg/m s rght Bonry qtons or φ lt n φ rght N.45N 33.55φ.55φ qtons III.45φ φ.55φ φ3 φ3.55φ φ4 φ4.55φ4.45φ5.45φ5. φ. φ φ3.45 φ4.45 φ 5 Solton on nt chrt or / omrson o Nmrcl n ct Rslts (/ /.) ct Nmrcl hng Ints Rclclt nt-rnc cocnts or.5 m/s s only chng.5 kg/m s,.5 kg/m s, / Solton on nt chrt φ 3.5 φ φ3.75 φ4.5 φ 5 36 M 69 omttonl l ynmcs 6

7 onvcton rvtvs brry 7, omrson o Nmrcl n ct Rslts (/ / 5) ct Nmrcl Rsolv Gr hng δ rom. to.5 hngs to kg/m s stys t.5 kg/m s / rto chngs to.5 In rst ml /. In scon ml / 5 / rto δ/ s cll th cll clt nmbr Rslts or δ.5 (/.5) on nt sl omrson o Nmrcl n ct Rslts ( /.5) ct Nmrcl ht s Hnng Hr? hn lo trm s lrg comr to son trm δ/ th s o vrg vls os not rcognz th rctonlty o th lo Kng th / rto smll llos s o vrg vls n convcton trms Rqrs smll δ or lrg lo rts Ho cn s lrgr δ or lrg lo rts n k ccrcy? M 69 omttonl l ynmcs 7

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