Complete the Online Evaluation of 2.29

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1 Complt th Ol Evaluato of Th sprg 2015 d-of-trm subjct valuatos opd o Tusday, May 5 at 9 am. tudts hav utl Moday, May 18 at 9 am to complt thr survys. Emal TA to gt two bous pots (survys ar aoymous Numrcal Flud Mchacs FJL Lctur 24, 1

2 REVIEW Lctur 23: Ft Elmt Mthods Itroducto Numrcal Flud Mchacs prg 2015 Lctur 24 Mthod of Wghtd Rsduals: Galrk, ubdoma ad Collocato Gral Approach to Ft Elmts: tps sttg-up ad solvg th dscrt FE systm Galrk Exampls 1D ad 2D Computatoal Galrk Mthods for DE: gral cas Varatos of MWR: summary Ft Elmts ad thr bass fuctos o local coordats (1D ad 2D Isoparamtrc ft lmts ad bass fuctos o local coordats (1D, 2D, tragular Hgh-Ordr: Motvato Cotuous ad Dscotuous Galrk FE mthods: CG vs. DG Hybrdzabl Dscotuous Galrk (HDG: Ma da ad xampl DG: Workd smpl xampl a ( x L ux ( f ( x Rx ( ux u( ( a ( x Lux ( f( x Rx ( 0 1 R( x w ( x dxdt 0, 1,2,..., Numrcal Flud Mchacs FJL Lctur 24, 2 tv

3 TODAY (Lctur 24: Ft Volum o Complx Gomtrs Turbult Flows ad thr Numrcal Modlg Ft Volum o Complx gomtrs Computato of covctv fluxs Computato of dffusv fluxs Commts o 3D Turbult Flows ad thr Numrcal Modlg roprts of Turbult Flows trrg ad Mxg Ergy Cascad ad cals Turbult Wavumbr pctrum ad cals Numrcal Mthods for Turbult Flows: Classfcato Drct Numrcal mulatos (DN for Turbult Flows Ryolds-avragd Navr-toks (RAN Larg-Eddy mulatos (LE Numrcal Flud Mchacs FJL Lctur 24, 3

4 Rfrcs ad Radg Assgmts Chaptr 8 o Complx Gomtrs of J. H. Frzgr ad M. rc, Computatoal Mthods for Flud Dyamcs. prgr, NY, 3rd dto, 2002 Chaptr 9 o Turbult Flows of J. H. Frzgr ad M. rc, Computatoal Mthods for Flud Dyamcs. prgr, NY, 3rd d., 2002 Chaptr 3 o Turbulc ad ts Modllg of H. Vrstg, W. Malalaskra, A Itroducto to Computatoal Flud Dyamcs: Th Ft Volum Mthod. rtc Hall, cod Edto. Chaptr 4 of I. M. Coh ad. K. Kudu. Flud Mchacs. Acadmc rss, Fourth Edto, 2008 Chaptr 3 o Turbulc Modls of T. Cbc, J.. hao, F. Kafyk ad E. Laurdau, Computatoal Flud Dyamcs for Egrs. prgr, 2005 Numrcal Flud Mchacs FJL Lctur 24, 4

5 Ft Volums o Complx gomtrs FD mthod (classc: Us structurd-grd trasformato (thr algbrac-trasft, gral, dffrtal or coformal mappg olv trasformd quatos o smpl orthogoal computatoal doma FV mthod: tarts from cosrvato qs. tgral form o CV d dt C CV C C CV dv ( v. da q da. s dv Advctv (covctv fluxs Othr trasports (dffuso, tc W hav s prcpls of FV dscrtzato Covctv/dffusv fluxs, from 1 st - 2 d ordr to hghr ordr dscrtzatos Ths prcpls ar dpdt of grd spcfcs, but, um of sourcs ad sks trms (ractos, tc vral w faturs ars wth o-orthogoal or arbtrary ustructurd grds Numrcal Flud Mchacs FJL Lctur 24, 5

6 Exprssg fluxs at th surfac basd o cll-avragd (odal valus: ummary of Two Approachs ad Boudary Codtos t-up of surfac/volum tgrals: 2 approachs (do thgs oppost ordr 1. ( Evaluat tgrals usg classc ruls (symbolc valuato; ( Th, to obta th ukow symbolc valus, trpolat basd o cll-avragd (odal valus ( F f da F ( ( F ( ( ( 's s ' s ( ( mlar for othr tgrals: 1 ( s dv, dv, tc V V V 2. ( lct shap of soluto wth CV (pcws approxmato; ( mpos volum costrats to xprss coffcts trms of odal valus; ad ( th tgrat. (ths approach was usd th xampls. ( a ( x a x (x ( ( a ( x ( x a x F V ( F f da Boudary codtos: ( ( 's s ( ( 's s mlar for hghr dmsos: ( xy, ( xy, ; a tc ( x, y ; tc a Drctly mposd for covctv fluxs O-sdd dffrcs for dffusv fluxs (From lctur 16 Numrcal Flud Mchacs FJL Lctur 24, 6

7 Approx. of urfac/volum Itgrals: Classc symbolc formulas urfac Itgrals 2D problms (1D surfac tgrals Mdpot rul (2 d ordr: Trapzod rul (2 d ordr: mpso s rul (4 th ordr: 3D problms (2D surfac tgrals Mdpot rul (2 d ordr: Hghr ordr mor complcatd to mplmt 3D Volum Itgrals: 2D/3D problms, Mdpot rul (2 d ordr: 2D, b-quadratc (4 th ordr, Cartsa: F f da ( f fs 2 F f da O( y 2 ( f 4 f fs 4 F f da O( y 6 F f da f O y z 1 s dv, dv V V V 2 2 (, s dv s V s V V (summary from Lctur 15 Imag by MIT OpCoursWar. 2 ( F f da f f O y f xy 16s 4ss 4s 4sw 4s ss ssw s sw 36 Numrcal Flud Mchacs FJL Lctur 24, 7 y y j+1 y j y j-1 j WW x NW W W w w sw N s x s NE E y E x -1 x x +1 Notato usd for a Cartsa 2D ad 3D grd. EE

8 FV: Approxmato of co ovctv fluxs For complx gomtrs, o oft uss th mdpot rul for th approxmato of surfac ad volum tgrals Cosdr frst th mass flux: =1: f 1 v. Aga, w cosdr o fac oly: ast sd of a 2D CV (sam approach appls to othr facs ad to ay CV shaps. Md-pot rul for mass flux: C ( v. d Advctv (covctv fluxs 2 m f 1 d f f O( ( v. y j NW w w W sw N η η ξ E s E x ξ Th ut ormal vctor to fac ad ts surfac x y ar dfd as: Hc, mass flux s: j( y y ( x x j s s whr ( ( x 2 y 2 y x y m v.( x ( ( v v.. j ( u v Imag by MIT OpCoursWar. Numrcal Flud Mchacs FJL Lctur 24, 8

9 FV: Approxmato of covctv fluxs, Cot d Mass Flux Th mass flux for th md-pot rul: x y m ( u v m ( What s th dffrc btw th Cartsa ad oorthogoal grd cass? I o-orthogoal cas, ormal to surfac has compots all drctos All vlocty compots thus cotrbut to th flux (ach compot s multpld by th projcto of oto th corrspodg axs y j W NW w w sw N η η ξ s E E ξ x Imag by MIT OpCoursWar. Numrcal Flud Mchacs FJL Lctur 24, 9

10 FV: Approxmato of covctv fluxs, Cot d Mass flux for md-pot rul: Covctv flux for ay trasportd Is usually computd aftr th mass flux. Usg th md-pot rul: F ( v. d f ( ( v. m How to obta?, us thr: whr = valu at ctr of cll fac A lar trpolato btw two ods o thr sd of fac (also 2 d ordr bcoms trapzodal rul Ft to a polyomal th vcty of th fac (pcws shap fucto Cosdratos for ustructurd grd: x y m ( u v m ( Bst comproms amog accuracy, gralty ad smplcty s usually: Lar trpolato ad md-pot rul Idd: facltats us of local grd rfmt, whch ca b usd to achv hghr accuracy at lowr cost tha hghr-ordr schms. Howvr, hghrordr FE or compact FD ar ow bg usd/dvlopd Numrcal Flud Mchacs FJL Lctur 24, 10

11 FV: Approxmato of dffusv fluxs For complx gomtrs, w ca stll us th mdpot rul Md-pot rul gvs: k C Dffusv Fluxs F k d f f O k d. d f f O( ( 2 (k.... d Hr, gradt ca b xprssd trms of global Cartsa coordats (x, y or local orthogoal coordats (, t y j η NW N η ξ w ξ E w W s sw E x Imag by MIT OpCoursWar. I 2D: j t x y t Thr ar may ways to approxmat th drvatv ormal to th cll fac or th gradt vctor at th cll ctr As always, two ma approachs: Approxmat surfac tgral, th trpolat pcfy shap fucto, costrats, th tgrat Numrcal Flud Mchacs FJL Lctur 24, 11

12 FV: Approxmato of dffusv fluxs, Cot d 1 If shap fucto (x, y s usd, wth md-pot rul, ths gvs: d x y x F ( k. k k x y x Ca b valuatd ad rlatvly asy to mplmt xplctly Implctly ca b hardr for hgh-ordr shap fct (x, y (mor cll volvd 2 Aothr way s to comput drvatvs at CV ctrs frst, th trpolat to cll facs (2 stps as for computg from y j W NW w sw η N η ξ s x E Imag by MIT OpCoursWar. E ξ O ca us avrags + Gauss Thorm locally Drvatv at ctr avrag drvatv ovr cll dv x CV Gauss thorm for / x (smlar for y drvatv: Numrcal Flud Mchacs FJL Lctur 24, 12 x CV dv x dv. d c x C 4facs c x c

13 FV: Approxmato of dffusv fluxs, Cot d y j Hc, th gradt at th CV ctr wth rspct to x s obtad by summg th products of ach c wth th projcto of ts cll surfac oto a pla ormal to x, ad th dvdg th sum by th CV volum x x x CV x dv dv dv 4 c facs For c w ca us th approxmato for th covctv fluxs W ca th trpolat to obta th gradt at th ctrs of cll facs For Cartsa grds ad lar trpolato, o rtrvs ctrd FD W NW w w sw η N η ξ s x E Imag by MIT OpCoursWar. E ξ Cll-ctr gradts ca also b approxmatd to 2 d ordr assumg a lar varato of locally: E. ( re r Thr ar as may such quatos as thr ar ghbors to th cll ctrd at d for last-squar vrsos (oly drvatvs D Issus wth ths approxmato ar oscllatory solutos ad larg computatoal stcls for mplct schms us dfrrd-corrcto approach Numrcal Flud Mchacs c c FJL Lctur 24, 13

14 FV: Approxmato of dffusv fluxs, Cot d Dfrrd-Corrcto Approach: y j Ida bhd dfrrd-corrcto s to dtfy possbl optos ad comb thm to rduc costs ad lmat u-dsrd bhavor. om optos: If w work local coordats (, t: If grd was clos to orthogoal Cartsa, usg CD: If trpolat th gradt at th cll ctr: W NW w w sw η N η ξ s x E Imag by MIT OpCoursWar. E ξ A fast oscllatory soluto dos t cotrbut to ths 3 rd hghrordr choc, but gradts at cll facs would th b larg => oscllatos do occur: Th obvous soluto: d F ( k. k E r r E 1 E W 1 EE 2 r r 2 r r E W EE mplct r1 xplct mplct (1 (2 (1 wll oscllat Nd to fd othr optos/soluto Numrcal Flud Mchacs FJL Lctur 24, 14 WW φ W W W φ φ E φ EE E EE Imag by MIT OpCoursWar. r (1 (2

15 FV: Approxmato of dffusv fluxs, Cot d Dfrrd-Corrcto Approach, Cot d (Muzafrja, 1994: y j If l coctg ods ad E s arly orthogoal to cll fac, drvatv w.r.t ca b approxmatd wth drvatv w.r.t ξ (as bfor: approxmato clos to 2 d ordr F k k k d E r E r If grd s ot orthogoal, th dfrrd corrcto trm should cota th dffrc btw th gradts th ad ξ drctos W NW w w sw η N η ξ s x E Imag by MIT OpCoursWar. E ξ d F k k E whr th frst trm s computd as: k k re r brackt trm s trpolatd from cll ctr gradts (thmslvs obtad from Gauss thorm Hc:. ad. old F k k old d E r E r Numrcal Flud Mchacs FJL Lctur 24, 15

16 FV: Approxmato of dffusv fluxs, Cot d Dfrrd-Corrcto Approach, Cot d (Muzafrja, 1994: I th formula: old F k k d E r E r Th dfrrd corrcto trm s (clos to zro wh grd (clos to orthogoal,.. ad ξ drctos ar th sam (clos to ach othr. y j W It maks th computato of drvatvs smpl (amouts to sums of ghbor valus, rcall that: old old trpolatd from, NW w sw w N η s x E Imag by MIT OpCoursWar. η ξ E ξ rvts oscllatos sc basd o sums of c, wth postv coffcts W rmad Cartsa coordats (o d to trasform coordats, w just d to kow th ormals & surfacs, whch s hady for complx turbult modls Numrcal Flud Mchacs th lattr gv by.g. x 4 c facs c x c dv FJL Lctur 24, 16

17 om commts o FV o complx gomtrs L -E dos ot always pass through th cll ctr d som updats that cas othrws, schm s of lowr ordr (.g. approxmato s ot scod ordr aymor chms ca b xtdd to 3D grds but som updats ca also b dd For xampl, cll facs ar ot always plaar, hardr 3D Block B w N R R s R ' 1 ' Block-Itrfac l+1 l Cll-fac ctr L Block A Cll vrtx Cll ctr l-1 Th trfac btw two blocks wth o-matchg grds. Collocatd arragmt of vlocty compots ad prssur o FD ad FV grds. NE E' E V 4 V 3 Block-structurd grds ad std grds also d spcal tratmt For xampl, matchg at boudars (usually trpolato ad avragg Numrcal Flud Mchacs 4 V 5 C 4 C 1 V 1 3 C 3 C 2 V 2 2 Cll volum ad surfac vctors for arbtrary cotrol volums. Imag by MIT OpCoursWar. FJL Lctur 24, 17

18 Turbult Flows ad thr Numrcal Modlg Most ral flows ar turbult (at som tm ad spac scals roprts of turbult flows Hghly ustady: vlocty at a pot appars radom Thr-dmsoal spac: stataous fld fluctuats rapdly, all thr dmsos (v f tm-avragd or spac-avragd fld s 2D om Dftos Esmbl avrags: avrag of a collcto of xprmts prformd udr dtcal codtos tatoary procss: statstcs dpdt of tm For a statoary procss, tm ad smbl avrags ar qual Acadmc rss. All rghts rsrvd. Ths cott s xcludd from our Cratv Commos lcs. For mor formato, s Thr turbult vlocty ralzatos a atmosphrc BL th morg (Kudu ad Coh, 2008 Numrcal Flud Mchacs FJL Lctur 24, 18

19 U8 Turbult Flows ad thr Numrcal Modlg roprts of turbult flows, Cot d Hghly olar (.g. hgh R Hgh vortcty: vortx strtchg s o of th ma mchasms to mata or cras th tsty of turbulc Hgh strrg: turbulc crass rat at whch cosrvd quatts ar strrd trrg: advcto procss by whch cosrvd quatts of dffrt valus ar brought cotact (swrl, foldg, tc Mxg: rrvrsbl molcular dffuso (dsspatv procss. Mxg crass f strrg s larg (bcaus strrg lads to larg 2 d ad hghr spatal drvatvs. Turbult dffuso: avragd ffcts of strrg modld as dffuso Charactrzd by Cohrt tructurs C ar oft spg,.. dds Turbulc: wd rag of dds sz, gral, wd rag of scals Istataous trfac 1 ~ δ δ Turbult flow a BL: Larg ddy has th sz of th BL thckss Numrcal Flud Mchacs Imag by MIT OpCoursWar. FJL Lctur 24, 19

Wladr s scrapbook trrg ad Mxg Wladr. tuds o th gral dvlopmt of moto a two-dmsoal dal flud. Tllus, 7:141 156, 1955. Hs umrcal soluto llustrats dffrtal advcto by a smpl vlocty fld.

20 Wladr s scrapbook trrg ad Mxg Wladr. tuds o th gral dvlopmt of moto a two-dmsoal dal flud. Tllus, 7: , Hs umrcal soluto llustrats dffrtal advcto by a smpl vlocty fld. A chckrboard pattr s dformd by a umrcal quasgostrophc barotropc flow whch modls atmosphrc flow at th 500mb lvl. Th tal straml pattr s show at th top. how blow ar dformd chck board pattrs at 6, 12, 24 ad 36 hours, rspctvly. Notc that ach squar of th chckrboard matas costat ara as t dforms (cosrvato of volum. Numrcal Flud Mchacs Wly. All rghts rsrvd. Ths cott s xcludd from our Cratv Commos lcs. For mor formato, s ourc: Fg. 2 from Wladr,. "tuds o th Gral Dvlopmt of Moto a Two-Dmsoal, Idal Flud." Tllus 7, o. 2 (1955. FJL Lctur 24, 20

21 Ergy Cascad ad cals Brtsh mtorologst Rcharso s famous quot: Bg whorls hav lttl whorls, Whch fd o thr vlocty, Ad lttl whorls hav lssr whorls, Ad so o to vscosty. η (Kolmogorov mcroscal Imag by MIT OpCoursWar. Dmsoal Aalyss ad cals (Tks ad Lumly, 1972, 1976 Largst ddy scals L, T, U: L/T= Dstac/tm ovr whch fluctuatos ar corrlatd ad U = larg ddy vlocty (usually all thr ar clos to scals of ma flow Vscous scals:, τ, u v = vscous lgth (Kolmogorov scal, tm ad vlocty scals Hypothss: rat of turbult rgy producto rat of vscous dsspato Lgth-scal rato: Tm-scal rato: Vlocty-scal rato: L O UL T 3/4 / ~ (R L RL 1/2 / ~ O(R L U u O 1/4 / v ~ (R L Numrcal Flud Mchacs R L = largst ddy R ~ R ma to 0.01 R ma FJL Lctur 24, 21

22 Turbult Wavumbr pctrum ad cals Turbult Ktc Ergy pctrum (K: I th rtal sub-rag, Kolmogorov argud by dmsoal aalyss that K (, A K K A 1.5 2/3 5/ foud to b uvrsal for turbult flows Turbult rgy dsspato Turb. rgy Turb. tm scal u u L 2 Komolgorov mcroscal: z of dds dpd o turb. dsspato ε ad vscosty 3 Dmsoal Aalyss: u L 3 1/4 2 ' ( 0 Numrcal Flud Mchacs FJL Lctur 24, 22 u K dk Acadmc rss. All rghts rsrvd. Ths cott s xcludd from our Cratv Commos lcs. For mor formato, s L (larg dds scal u U u'

23 Numrcal Mthods for Turbult Flows rmary approach (usd to b s xprmtal Numrcal Mthods classfd to mthods basd o: 1 Corrlatos: usful mostly for 1D problms,.g.: Moody chart or frcto factor rlatos for turbult pp flows, Nusslt umbr for hat trasfr as a fucto of R ad r, tc. 2 Itgral quatos: Itgrat DEs (N qs. o or mor spatal coordats olv usg ODE schms (tm-marchg 3 Avragd quatos Avragd ovr tm or ovr a (hypothtcal smbl of ralzatos Oft dcompostos to ma ad fluctuatos: Rqur closur modls ad lad to a st of DEs: Ryolds-avragd Navr-toks (RAN qs. O-(spatal pot closur mthods Numrcal Flud Mchacs f f(r, Nu (R,r, Ra u u u'; FJL Lctur 24, 23

24 Numrcal Mthods for Turbult Flows Numrcal Mthods classfcato, Cot d: 4 Larg-Eddy mulatos (LE olvs for th largst scals of motos of th flow Oly approxmats or paramtrzs th small scal motos Comproms btw RAN ad DN 5 Drct Numrcal mulatos (DN olvs for all scals of motos of th turbult flow (full Navr- toks Th mthods 1 to 5 mak lss ad lss approxmatos, but computatoal tm crass from 1 to 5. Cosrvato DEs ar solvd as for lamar flows: major challg s th much wdr rag of scals (of motos, hat trasfr, tc Numrcal Flud Mchacs FJL Lctur 24, 24

25 MIT OpCoursWar Numrcal Flud Mchacs prg 2015 For formato about ctg ths matrals or our Trms of Us, vst:

2.29 Numerical Fluid Mechanics Fall 2011 Lecture 23

2.29 Numerical Fluid Mechanics Fall 2011 Lecture 23 2.29 Numrcal Flud Mchacs Fall 2011 Lctur 23 REVIEW Lcturs 22: Compl Gomtrs Grd Grato Basc cocpts ad structurd grds trtchd grds Algbrac mthods Gral coordat trasformato Dffrtal quato mthods Coformal mappg