Environmental Fluid Mechanics
Environmental Fluid Mechanics CVEN 5313
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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.
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Date Created: 10/29/15
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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
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