Navier-Stokes Eqns. υ ρ. Conservative form momentum eqn; x-component only. u x. u y. p z. u x. uv y

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1 3 Spaial Averaging 3-D eqaions of moion Saling > simplifiaions Spaial averaging Shear dispersion Magnides/ime sales of diffsion/dispersion amples

2 Navier-Sokes qns Conservaive form momenm eqn; -omponen onl 1 p fv w v υ ρ sorage or loal aeleraion adveive aeleraion 3 Coriolis aeleraion [f Coriolis param ωsinθ, θ laide] 4 pressre gradien 5 visos sress

3 Trblen Renolds qns [ ] ' ' ' ' ' 1 ' ' ' average ime e, ' w v p fv w v w v υ υ υ ρ Inser & ime average over bar Conini e.g., -omponen; erm -momenm 1 a b

4 Speifi erms Term a: old sbra imes onini eqn w v w v NC form of momenm eqn erm Terms 5 b w w w v v v ε τ υ ε τ υ ε τ υ ' ' ' ' ' ' ' ' ' ' -omp of rblen shear sress ε, ε, ε are rblen edd kinemai visosiies resling from losre model like,,

5 Speifi erms, on d Similar eqns for and eep has gravi. For nearl horional flow w ~ 0, -mom > hdrosai 1 0 ρ p p g 4 6 η p a ρgd Use in and eqns η,,

6 Speifi erms, on d Term 4 gd g p gd p p s a a η η ρ ρ η ρ η,, 4a 4b 4 4a amospheri pressre gradien ofen negligible 4b baroropi pressre gradien baroropi > ρ ρ s ons 4 barolini pressre gradien barolini > densi gradiens; ofen negligible

7 Simplifiaion Negle ε ; ε ε h ; ε ε Negle all pressre erms eep baroropi And drop over bars g fv w v ε h ε η 1 a 3 4b b1 b Conras wih mass ranspor eqn ± r w v 1 a b1 b 7 Pressre gradien 4 in momenm eqn > visosi b no alwas imporan o balane adveion a; depends on shear separaion. For mass ranspor, diffsivi b alwas needed o balane adveion a

8 Commens 3D models inlde onini hree omponens of momenm eq ma be hdrosai appro n mass ranspor eqns Above are primiive eqs, v, w; someimes differen form, b phsis shold be same Someimes frher simplifiaions Spaial averaging > reded dimensions

9 Frher possible simplifiaions Negle erms a and b1 fv g η Also negle erm 1 η fv g ε Also negle erm b η fv g ε Linear shallow waer wave eqn Sead kman flow Geosrophi flow

10 Spaial averaging 3-D eqaions φ,,, oean -verial -laeral -longidinal -D verial average φ,, shallow oasal; esar -D laeral average φ,, long reservoir; deep esar/fjord 1-D verial & laeral average φ, river; narrow/shallow esar 1-D horional average φ, deep lake/reservoir oean

11 Commens Models of reded dimension ahieved b spaial averaging or dire formlaion advanages of boh Demonsraion of verial averaging inegrae over deph hen divide b deph, leading o D deph-averaged models Disssion of ross-seional averaging river models

12 Verial Inegraion > D deph-averaged eqns,, h,, η,,,,,,,,,,,,,,,,,, U h noes se

13 Verial Inegraion, on d η -h η 1,,,,, d h In analog wih Renolds averaging, deompose veloiies and onenraions ino,, e. and spaiall average

14 Deph-averaged eqns Conini 0 v η v w, of b kinemai srfae from Mass and Momenm sraigh forward eep for NL erms d d d d d d h h η η momenm dispersion mass dispersion

15 Deph-averaged eqns, on d -momenm ρ τ ρ τ ε ε η b s T L v g fv v v ε ε Deph-ave long. diff Long. dispersion

16 Deph inegraed eqns, on d Mass Transpor b s T L v v Deph-ave long. diff Long dispersion

17 Commens ε L, ε T, L, T are longidinal and ransverse momenm and mass shear visosi/dispersion oeffiiens. ε L >> ε T and L >> T, b relaive imporane depends on longidinal gradiens Dispersion proess represened as Fikian eplained shorl

18 Bondar Condiions: momenm τ s ρ C D U w V w U w Srfae shear sress de o wind omponens U w and V w ; eernal inp nless opled air-waer model Assmes U w >> s ; C D drag oeffiien ~ 10-3 more in Ch 8 τ b ρ C f v Boom shear sress ased b flow omped b model Differen models for C f Dar- Weisbah f; Manning n; Che C, τ f ρ 8 e.g., b

19 Bondar Condiions: mass ranspor s Srfae mass ransfer air-waer ehange s 0 No fl de, sal s Sore DO > 0 s < 0 Sink VOC

20 No fl de, sal Sore pore waer diffsion Sink rae meals bond b anoi sedimens Bondar Condiions: mass ranspor b Benhi mass ransfer sedimen-waer ehange b 0 b > 0 b < 0

21 Magnide of erms: f f L b , 8 / veloi shear ~ ' ~ > ρ τ S A radis hdrali p A R gs Sg R p ASg p p F Sg A F b f g / ; ; ρ τ ρ Normal flow; gravi balanes friion

22 on d 0.07 Seen previosl; from analog of mass and momenm onservaion Renolds analog and log profile for veloi τ vm vm τ vm 7 ; if 0 vm 150

23 Transverse miing: T T sa 0.15 Laboraor reanglar hannels T sa 0.6 Real hannels irreglariies, braiding, seondar irlaion τ m m 0.5B T 0.5B B ; if B 0 B hannel widh

24 ample B 100 m, 5m, 1 m/s X vm m m 17B / /5 34,000 m 34 km I ma ake qie a while before onenraions an be onsidered laerall ransversall niform

25 Simplifiaions Sead sae; deph-averaged; no laeral adveion or long dispersion; no bondar fles b s T L v T Simple diffsion eqaion: solions for oninos sore a 0 T Uniform hannel ons B

26 1.0 B q d - dimensionless mlaive disharge d oll and Jirka 1986 B m d - dimensionless onenraion q d 0 q σ d - dimensionless longidinal disane T B T B Noe: m 0.5B Figre b MIT OCW. T

27 A sefl eension: mlaive disharge approah Yoskra & Sare, 1976 Use mlaive disharge Q insead of as laeral variable Q Q ' ' d' 0 Q Q T Q Q D D behaves mahemaiall like diffsion oeffiien, b has differen dimensions; an be approimaed as onsan ross-seional average: Q 1 T dq Q > D D 0 Q Can se previos analsis

28 Longidinal Shear Dispersion Wh is longidinal dispersion Fikian? Original analsis b Talor 1953, 1954 for flow in pipes; following for D flow afer lder 1959

29 Longidinal Shear dispersion Wh is longidinal dispersion Fikian? Original analsis b Talor 1953, 1954 for flow in pipes; following for D flow afer lder 1959 o 1 L

30 Shear dispersion, on d o 1 L ; ; ; ζ ζ ζ ζ ζ ζ τ τ τ ζ

31 Shear dispersion, on d ζ ζ ζ ζ ζ ζ τ τ o 1 L ζ τ ζ ζ << > >> << > >> << > >> verial dispersion longidinal L b L a 5, 6 << 7 4 << 3 1, << 3

32 Shear dispersion, on d o 1 ζ 3 7 We wan L Differenial adveion balaned b ransverse diffsion and o show ha i is ~ L Inegrae over wie o ge ; mlipl b ; inegrae again and divide b deph average; add mins sign

33 Shear dispersion, on d 1 0 d 1 ζ ddd L!!! L Ih I h dimensionless riple inegraion ~ 0.07 ~ ; ~ L 5.9 L 10 r o lder 1959 sing log profile for Talor 1954 rblen pipe flow r o radis Use L o ompe σ L or se measred σ L o dede L

34 Commens L involves differenial adveion wih ransverse miing in direion of adveion gradien L ~ 1/ ; perhaps oner-iniive, b look a ime sales: L ~ T U Reall Talor s Theorem D ~ R τ dτ 0

35 A hogh eperimen Consider he rip on he Mass Trnpike from Boson o he NY border ~150 miles. Assme wo lanes in eah direion, and ha ars in lef lane alwas ravel 65 mph, while hose in he righ lane ravel 55 mph. A he sar 50 ars in eah lane have heir ops pained red and a helioper observes he dispersion in heir posiion as he ravel o NY 1 ow does his dispersion depend on he freqen of lane hanges? Wold dispersion inrease or derease if here were a hird middle lane where ars raveled a 60 mph?

36 Thogh eperimen, on d L ~ Wha are he analogs of, and

37 Thogh eperimen, on d ~ nmber of lanes L ~ ~ freqen of lane hanging ~ differene beween average and lane-speifi speed 1 Dereasing inreases dispersion as long as here is some Inreasing lanes inreases, dereases mean sqare, / lanes 50 / 3 3 lanes If is onsan, ne effe is inrease in L b 3/ /3 50%

38 1D river dispersion A da A 1 ; Inser ino G and spaial average Conini q L A A q L laeral inflow/lengh [L /T] Mass Conservaion onservaive form L L i q Ar A A A A A L Longidinal dispersion again

39 1D river dispersion, on d NC form from onservaive eqaion mins imes onini L L i q Ar A A A A L q A A Mass Conservaion NC form A q r A A L L i L 1 Noe: if L >, inreases; if L <, dereases dilion

40 L for rivers lder formla aons for verial shear OK for deph averaged models ha resolve laeral shear; here we need o parameerie laeral and verial shear. Analsis b Fisher 1967 I b B T B river widh T ransverse dispersion oeffiien I b ND riple inegraion aross A ~ 0.07 L Same form as lder, b now ime sale is B / T, raher han /. T >, b B >> > his L is generall mh larger

41 L for rivers, on d Using approimaions for, T, e. or if 0.01 L L 0 B B 0.01 Fisher 1967; sefl for reasonabl sraigh, niform rivers and hannels L B 4

42 Magnide of erms, revisied B r L o L L T Verial Diffsion Transverse Diffsion in Channels Longidinal Dispersion deph-averaged flow Longidinal Dispersion rblen pipe flow Longidinal Dispersion rivers

43 Previos ample, revisied B 100 m, 5m, 1 m/s, L 0.01 B m /s d σ d L > Gassian Disribion; b onl afer ross-seional miing m 0.5B m T m 34 km if poin sore on river bank; B

44 Previos ample, revisied B 100 m, 5m, 1 m/s, L 0.01 B m /s d σ d L > Gassian Disribion; b onl afer ross-seional miing m 0.5B T m 4 imes less if poin sore in mid-sream B

45 Previos ample, revisied B 100 m, 5m, 1 m/s, L 0.01 B m /s d σ d L > Gassian Disribion; b onl afer ross-seional miing m 0.5B m T Less sill if disribed aross hannel b no ero B

46 Sorage ones Real hannels ofen have bakwaer sorage ones ha inrease dispersion and give long ails o disribion A A s,

47 Sorage ones, on d 1 d d s A α A s 1 A A s L q L A L α s Main hannel Sorage one A ross-seional area of main hannel A s ross-seional area of sorage one α sorage one oeffiien rae, -1 ; like q L /A If o mlipl 1 b A and b A s, he ehange erms are αa s - and αa s - Reall he same proess as longidinal dispersion, b insead of ars in eiher fas or slow lane, some are in he res sop.