New User Special Price Expires in

Let's log you in.

Sign in with Facebook


Don't have a StudySoup account? Create one here!


Create a StudySoup account

Be part of our community, it's free to join!

Sign up with Facebook


Create your account
By creating an account you agree to StudySoup's terms and conditions and privacy policy

Already have a StudySoup account? Login here

Environmental Fluid Mechanics

by: Lina Vandervort

Environmental Fluid Mechanics CVEN 5313

Lina Vandervort

GPA 3.72

John Crimaldi

Almost Ready


These notes were just uploaded, and will be ready to view shortly.

Purchase these notes here, or revisit this page.

Either way, we'll remind you when they're ready :)

Preview These Notes for FREE

Get a free preview of these Notes, just enter your email below.

Unlock Preview
Unlock Preview

Preview these materials now for free

Why put in your email? Get access to more of this material and other relevant free materials for your school

View Preview

About this Document

John Crimaldi
Class Notes
25 ?




Popular in Course

Popular in Civil Engineering

This 30 page Class Notes was uploaded by Lina Vandervort on Thursday October 29, 2015. The Class Notes belongs to CVEN 5313 at University of Colorado at Boulder taught by John Crimaldi in Fall. Since its upload, it has received 29 views. For similar materials see /class/231886/cven-5313-university-of-colorado-at-boulder in Civil Engineering at University of Colorado at Boulder.


Reviews for Environmental Fluid Mechanics


Report this Material


What is Karma?


Karma is the currency of StudySoup.

You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 10/29/15
VI VOR39UQTY AND CARCULR TkON 39 Vorhcijrtl LS Vx Circulajaon T j 73 1 a 570mg W 96 x W HM 0 p VOV MU Jr 1 my mm s 5 W3 ILJDU HLS young cqu bow ch03 sur ng So Vor kcjl x is 39H xe dumbhon PU Unnij arem J m 39 6 i AcQNA ai 6k POM L g 1 a 4 NEH 39H39mfk T 15 c SCL QF bu JO 5 m VCLsrbr 9quot T hf Amouth of VOr HcHILr in S H V2 VOWEL 15 A measure a Hm 50N 9oq rojmjcmn 0d an Fo mi 1 A wwuwhwsw 32ltgtu3 9w 9 ax ax3 3x3 3quot u 2x 3 RCCCLH 5 O 50 0 VM 2 Si iv 3 S I 3 1 J J 3 03 O a 5 32 0 0 Mtwanjuimr 5 B R S Hains BY Aegnijcmh VOf I gE is Junk Hm aver548 YOOCHDYXQ1 rude O c a QAemen39h PR1de VarJriciw Cam Ba 51m Newark Fiac ms 3 obsefvimj a 5 mam 3an 0 angular 3t exi n 3 is a Tolfahb No Vo cijnr Vorjrialnf Emmahon 13er 39139 HM Jr werg m3 u 0 W Fonz VORTEX lt53 V Jr ih d FMquot 2 quotPage 3 R z r us 9 quot 17 No Smam Uw 0 J mlmjriow rake 1 u2 O re a plane A 8 T n a 8 mega Sheamhnw N e mix Jane Quick elemenj rolccdres oJoouL L i k s cu 39er 3 Vor ini3c7 1 i 3amp1 8amp9 F amp 03g quot quot 0 50 cm on 139 33 21 A vgnixx B 935 all 1 we at S O t 1 2 L a 1 w A g 139 c n W 4 roji39al on Hula va Ca wldrc HM Circukaho 39Co Um Erect Varkm m a Circle wi x Fa lus R con rm Q 0 02 m m T 135 Edema BQDK1A6 My 0 D 7 7 Chaddion cranes as R gt mcor wlfh arm D Vothci Cifcwhfkon ch umk Mcu is CV6 W fe LEI1 R EXAMPLE 7 FREE VORTEX 1am Vovaexgt z T URE STRAIN u C e 1quot 9 Ho SEAR Ar o a O S rmmhvxg The glu39tc damen Ud i vsocs sroimi03 J 0qu ho 1 DATDA IOH 3 no Vorjn39ciJrr ELLe Du A H B r r 29 22 O Ques l on 99 quot 3 333 O Mud is 25 9239 3r QJC r O i 3 3m 02 F 94m i at O Catcx q kc Hm Circuh mn FO 6 Free Vorjttx 39m a Circla wiik radius R can rcmcl 05k Tquot O a 2V T u39d jKampe ZTC O Nho r a 30kg on Wk 5 T i O 2 RECku Joke Sin ulmri39l39r r30 CoJCu ock he Circdquion gov Jme rejion S belowJ 5 1085 no r Conhm m omjm 03 re innev Cirdc 392 Qk39 r n h m hon hiHon ow ur cirde r A B D C E D W C AB C j E Ru g j Kl 0 uni3 lt0 on DA 7 A Vorku 39 ZWTC O is ad We orfjin Some 0 Vorhces Ax RANKILXE VORTEX Ideahatl B L W m u fcrjrex gt 1 R E blEar 39 Cr DR i 9 y z a 3 R QUE E Bgt Bamroe VORTEX Consickr CL 100 fitmensfomql 7 UL YomPreSBL 7 Via 0 gt g STREAMFUNL nomg 2D 90 aimansrona inwmprtssilo i owquot ulxtxl 2XI5XL 3 0 3 0 Ex Bx bxth H 6 con clichovxs we Com Jc nt 0V sjrrmmgundion W 20 X1 XL SMCM quotzxanC 2 39 n F U ax ax I A ow ic nci a SJWCAM39CunCJ On wiil alum Sad39ftsf ll CDHHWLLHY i jilt i 9 7 QXI 9x1 3quot XL 9 axl 9W 9W axlbxl 3x 9x1 Two imimAa Pngat x ts 0amp1 3teqmgmnc oma G W Cuw Awa bd Hm limdibvx wzcons mk is O SAVx ethmc in 8va 43b 0 g1 xproog We at Ch aw Ax gylxj Bx 3 9X 2 um 252 u 933 ax f 5 So mlampamp uldxl I Lu c0mm Wm gt Liviaiulclxl 0quot LLL ix LN dx 4 f EAOFL 9 Idogi Vcdtw l7 4 Cohsan 5 0Ft 0 4 CO 539 lt71quot Cons SF3 vohma TEE Liowrock Karl wean mo 539 camliv lw 39U 1L and 5039 4 3 3 WE PA wilch 5 MA 11 OUR comsltm5 me A M E 3 1 39ix 1r i om geome HY Q12 CUZ 2amp5 ampX j mu Mu 5 PMS j u b z Vicki AI 3 is Lb B u gram Frzw ous Page m A j CW 2 506 WA H39B 50 QM 4 3 quot q A 39 wants 63 Fac m 3n HM 1i39e k 3h 0399 Mann q 1 quotUna Ic ammpk 1 W SH Gin P K m a quot 34 M xl 9mm Ugr ex x 2quot gig 39xl Chad Conthui 1 23134 f 333 1O axi axl PIM39 feamhnts39 Coda 45 X HO 2 1 LhYPtrbokS oR Use cmhufimj on L uxl as shown xsizesS xstepz02 ysizetb ystepa02 contourranqe2A contourstep4 xy meshgridthsizecxstepxsizeyysizeyscepysxze3 create 2D grid 93inxY calculate Psi an gtid vlt ontourrangecontour2tepcontcuxzangea define contour levels ChtcontourXYPsiv k39 3 make the contour plot clabelch label the contours made in the previous step axis equal make the x and y axes anve the same scale axiattxaize xsize ysize ysize 3 secs the axis range Q1 R quot K L z EXQDOPampIFZ Fbu Odovmcl a CANNAQY r l LP Uc rsme l 1 ND wd ArT 85 3 Phi sremmhnes m cax resfan CoerciinoA es r Xlri39 r 9V Sm 6 X quotJ WC le XL rt 50 Hp U 1 1 gt k x K D Xi1x X Z Y Z L quot Upsquot conbvfh 3 OH xslze101 xstepn02 ysize101 ystep02 contourrange10 contourscep1 0010 r3phere025 xY a meshgrid xsizeXSCepxsizeysizeystepysize create 2D grid THETARHO catt2p01XY iconvert to polar coords PsiUORHO39sinTHETA1rphereRHO 2 calculate Psi on polar grid v concourrangecontouzstepcontourtange define contour levels contourXYPsiv39k39 plot contours on Cartesian grid axis equal axis xsize xsize ysize ysizel mo 06 V 39 a4 POT ENTUAL Roms Irrolrcthonql ows V xii 05 arc cam I39eretl PDJrean ows bccauSL Huir VclociJrY giclLs Cam b6 UPTCSSCA in erm5 oquot a Valoq Ego01qu 1 0L SCUM und39ion R Show HMS Canada a 4 00quot dimenss ond holm ovqu 10M 1 u 1o Sm VXQO wt have 9W 9w O axx T warv Hm masks L Samkar uncion Ae m Suck nevi LA 39 Q1 34gt axz Th5 Ml aulromcdicdw 50th V x39Cg a O Idog Po mR d5 an Eddml 5 5 ch FuncJ ovxs Via 39L u ax 9x1 9 9 i Q 9X 3x 1 I F 15 dun 90m Hal39s and CiulPoknkml fms Cgt are Q39Fenampicu qr 0 Sf cxmvreg 4 Q EWE Fina Hm Vdoci Folf j ovl Corrcsfomlinj Jr0 We 5remunc an XIXZ gt UH 29 xr Soluhon 1 Docs L wha Fo i mkd 003 V39K 3 auL 9U k ll DXI ex 0 30 CNS15 245 J ax in Xt gt Jgtgclxl 1581 Hux DqS 3x1 quot M m 7 lt5 5m 4x 391 U MI W mm xf x3 5mm wk rejmirumcm 5 SUMmmN FOR STRERMFonxoME Vennng POTENTmL STREAMFUNCTIDN w 6 EX1S5 06 a wo39ampImensa oma3 mtomfxasw ut How 0 9 Con an E5 5 reamiinc 9amp5 9 5 quot11 2 75 Vdmdric Howrd c VELOCITY 39POTEN39WAL gz 39 Exib k or 1U irroxLaJ ova ows 9 Cam be exf39cnaccl JIt 3 9 A 9 5 Cons hn39 1 FchM 1qu 1 Us a I mom is H Xuicn v xigtrc re 0 than Wt 3325 leads 0 9 ags 0 VL O Ldplaxx EUaio 39INTEGRAL meOREMS E KMULP S oF GAuss AND STOKES R664 UM gun umenkcd theorcm 0 cdcuhs G Nm a unc ion dx7 b L If ax j rifwx ltb M a A Where a CmJ b are ConS ltan tS 00 The 39mHjm o d ova MM Con c39tmous range a S X i 3 Can be evduoH39eA i Calcwlq Hnj j dx only 0L1 Una enampEoins 0 he romje The 1Com le a okes omcl Gauss am omalajous 0 Hm gunAAmen39ka hemem 0 CQ CQ U s QXCQ Pt in huo am che chmensionql domains STOKES THEOREM relaJces Hag sur gxce mh rak 0 Hag Cu 0 a Ved39or 0 at line m rejrol 09 the Vccor a on 39Fhe closecl Curvc boundinj UM SawQue j VXV I 2 D Surth n q A s A V nip 1 L Note W nem V is VdoLHY nen 33 39 hV39 19 T Circulation Vow hd cv 7x7 lw GAus s39 THEOREM rcla ces Hm Volum quotnjrcjml O39C the gradienk or Aivarsencz 039 m Ved br ov Qnsor I D Hgg supgtlce 1er 0 HM V COf Or CHSOF On kke SurFacg bowndws er V0ume 3L 5D V0um L at End W LNLTJK d5 amaze H 0Q 5 9i T11 W j n Tn AS Amrjwce R s This 5 Mae 30 Cchi Diver mq mcorcm 381 W w 38139 WM i Una rcsion oC n 39 3r ion s changins 0L6 Ox vmcion a 39Hme LElBNT2 395 THEOREM ONE DmeNsloNAL IEKsmN HQ b A cl b lt1 E x w AX S 39 ch a Hb E Hal c a Lt 0quot H 3quot Hint M i Ht gt ib iifck E Lt 44 I am LLB l mm Mu At h No h 1 Q at and than 31 Jb x 3 AX 2 j tit ix 0 THREE DlMENSIONAL VERSAON EHO j T39LJ XM W 3 W SMWK TL AS M v S W is H w Veloci w 04 Pm Sudan 0quot V gels 5 Mom 4 m bound damJc mm m on mt an di cren HA39Hon mm l iH erxhon is Ih h rclnanacagbte An Awhcahon 01C We H cOrcms of Gauss and Leibni rz DcrivaHon 039 Una Tumsch Wcorem Consular a Malcricd Ioww Hrer C0n5i of a 1er Sci of nial FAAidas amA whose bounclinj bureau movcs wi dq Une Quilt 1 c unk Con l39aihs some SCoaf alwmjcibl 5 83 had 50 km he oal amounJc 0 m kc Mdcraal Vo ume V is 39 J 975 H Wt Lhc Baunaqn of HM Mo39eria Vo wmt is hmc JePchm t ARA 42k 1mm rake 0 Change 0 C5 M H1 VOUVYIL 5 A a w VG We Cam Isl n3 Hm Jerivwhvz insidn Hne vHejro usins Ltilom39 d M 3 d j w j at amp v f n u S as M s Sur cace moves wika Rubi which has Jdoc vY Ii Now Fewr HM UK LLQ i f 3 ol am a Vo uwu inhavml USMj 6041532 jfxuqsds 3v M CW 5 V 16 Combgnmg Gums he Trunsv mrt Theorem d A v v V cu 525 d o at WA 41 W6 V o cc V39 Cal I 3 Lki LL 3 9533 we wnl Show had gov incompresser HuiJs V39 I BLUL O n which Case V so ij clv 3 M wv cW39 Vm v Furikermow or o conservq cive scdar 2 Lt H MS 3 aquot Ci V9 CW 3 O 39For any OrH HMY W 3 V and he C2 in co m FW I O Conseruwk wc Scdar 1 qu Consicler o one Aimcws onad mow 292 Be at quotL U 3x O kme radie 0 Chan c Chomjc in clue 0 9 4 0 Poml JV who H WOUKj W 0k O Hou f Vuth 91ch EXmmEet 92 quot A 3932 at 9x Jcime roam owc dwwnje oxc Cd Fo w P y UL 3t 5 o Pll mere 4 39 55 39UL E gt 0 so gt O A Primer on Index Notation John Crimaldi August 23 2007 1 Index versus Vector Notation Index notation aka Cartesian notation is a powerful tool for manip ulating multidimensional equations However there are times when the more conventional vector notation is more useful It is therefore impor tant to be able to easily convert back and forth between the two This primer will use both index and vector formulations and will adhere to the notation conventions summarized below Vector Index Notation Notation scalar a a vector 6 a tensor A Aij In either notation we tend to group quantities into one of three categories scalar A magnitude that does not change with a rotation of axes vector Associates a scalar with a direction tensor Associates a vector or tensor with a direction E0 Free Indices a A free index appears once and only once within each additive term in an expression In the equation below i is a free index ai eijkbjck DijEj b A free index implies three distinct equations That is the free index sequentially assumes the values I 2 and 3 Thus a1 171 61 aj bj cj implies a2 122 Cg as 1 3 63 c The same letter must be used for the free index in every additive termi The free index may be renamed if and only if it is renamed in every termi d Terms in an expression may have more than one free index so long as the indices are distinct For example the vectornotation expres sion A ET is Written Aiv BijT B 391 in index notationi This exprgsiorimplies nine distinct equations since i andj are both free n ices e The number of free indices in a term equals the rank of the term Notation Rank scalar a vector ai l tensor Aij 2 tensor Aijk 3 Technically a scalar is a tensor With rank 0 and a vector is a tensor of rank 1 Tensors may assume a rank of any integer greater than or equal to zero You may only sum together terms With equal ranki f The rst free index in a term corresponds to the row and the second corresponds to the column Thus a vector Which has only one free index is Written as a column of three rows a1 6 ai a2 as and a rank2 tensor is Written as A11 A12 A13 Aij A21 A22 A23 A31 A32 A33 3 Dummy Indices a A dummy index appears twice Within an additive term of an expres sion In the equation below j and k are both dummy indices ai eijkbjck Dijej b A dummy index implies a summation over the range of the index an E all a22 ass c A dummy index may be renamed to any letter not currently being used as a free index or already in use as another dummy index pair in that term The dummy index is local to an individual additive term It may be renamed in one term so long as the renaming doesn7t con ict with other indices and it does not need to be renamed in other terms and in fact may not necessarily even be present in other terms 4 The Kronecker Delta The Kronecker delta is a rank2 symmetric tensor de ned as follows 6H7 1ifz j 11 0 ifiy j or 100 6010 001 5 The Alternating Unit Tensor a The alternating unit tensor is a rank3 antisymmetric tensor de ned follows 1 if ijk 123 231 or 312 eijk 0 if any two indices are the same if ijk 132 213 or 321 H The alternating unit tensor is positive when the indices assume any clockwise cyclical progression as shown in the gure m b The following identity is extremely useful EijkEilm jl km 5jm5kl 6 Commutation and Association in Vector and Index Notation a In general operations in vector notation do not have commutative or associative properties For example a X E y E X a b All of the terms in index notation are scalars although the term may represent multiple scalars in multiple equations and only mul tiplicationdivision and additionsubtraction operations are de ned Therefore commutative and associative properties hold Thus aibj bjai and aibjck aibjck A caveat to the commutative property is that calculus operators discussed later are not in general commutative 7 Vector Operations using Index Notation a Multiplication of a vector by a scalar Vector Notation lndex Notation abE abici The index i is a free index in this case b Scalar product of two vectors aka dot or inner product Vector Notation lndex Notation E I c aibi c The index i is a dummy index in this case The term scalar prod uct77 refers to the fact that the result is a scalar c Scalar product of two tensors aka inner or dot product Vector Notation lndex Notation c Aiiji c The two dots in the vector notation indicate that both indices are to be summed Again the result is a scalar d Tensor product of two vectors aka dyadic product Vector Notation lndex Notation BE g aibj Cij The term tensor product77 refers to the fact that the result is a ten sor e Tensor product of two tensors Vector Notation lndex Notation AEQ AiijkCi The single dot refers to the fact that only the inner index is to be summed Note that this is not an inner product f Vector product of a tensor and a vector Vector Notation lndex Notation E E aiBlj0j Given a unit vector 73 we can form the vector product ft E E In the language of the de nition of a tensor7 we say here thatThen ten sor E associates the vector Ewith the direction given by the vector 73 Ailso7 note that E E f E E g Cross product of two vectors Vector Notation lndex Notation E X b E eijkajbk Ci Recall that a17a27a3 X 11171727 113 2173 a31727 a3171 a11737 a1172 a2111 Now7 note that the notation eijkajbk represents three terrns7 the rst of Which is Eijkajbk 7 Eiikaibk 512ka2bk Elskasbk Eiiiaibi 5112041172 5113041173 5121042171 6122a2b2 6123a2bs Eisiasbi 6132asb2 Eissasbs 6123a2bs 6132asb2 12123 7 13122 h Contraction or Trace of a tensor sum of diagonal terms Vector Notation lndex Notation tr b Aii b 8 Calculus Operations using Index Notation Note The spatial coordinates z y72 are renamed as follows I A 11 y A 12 2 A 13 a Temporal derivative of a scalar eld 4511 12 13 t 19 i E E at W There is no physical signi cance to the 0 subscripti Other notation may be used b Gradient spatial derivatives of a scalar eld 41511 12 13 t 19 T11 61 19 T12 62 19 T13 63 These three equations can be Written collectively as 19 7 811 7 ln vector notation 8145 is Written V45 or grad Note that 8145 is a vector rankli Some equivalent notations for 8145 are 8145 E BIZ45 91 and occasionally 91115 E 19157139 c Gradient spatial derivatives of a vector eld 511 12 13 t 85 Th Lilla BE i612 920 85 T Egal These three equations can be Written collectively as 8a azj E Bj al In vector notation Bjai is Written V5 or grad 6 Note that Bjai is a tensor rank2 alal hag ag at 1quot grad E a V Bjai 92m 92 92 I 83 agag Bgug The index on the denominator of the derivative is the row indexi Note that the gradient increases by one the rank of the expression on Which it operates d Divergence of a vector eld 511 12 13 t divEVE8iaib Notice that Biai is a scalar rank0i Important note The divergence decreases by one the rank of the expression on Which it operates by one It is not possible to take the divergence of a scalar


Buy Material

Are you sure you want to buy this material for

25 Karma

Buy Material

BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.


You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

Why people love StudySoup

Jim McGreen Ohio University

"Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

Amaris Trozzo George Washington University

"I made $350 in just two days after posting my first study guide."

Bentley McCaw University of Florida

"I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

Parker Thompson 500 Startups

"It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

Become an Elite Notetaker and start selling your notes online!

Refund Policy


All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email


StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here:

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.