# Chapter 4: Fluid Kinematics

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1 4-1 Lagangian g and Euleian Desciptions 4-2 Fundamentals of Flow Visualization 4-3 Kinematic Desciption 4-4 Reynolds Tanspot Theoem (RTT)

2 4-1 Lagangian and Euleian Desciptions (1) Lagangian desciption of fluid flow tacks the position i and velocity of findividual id paticles. Based upon Newton's laws of motion. Difficult to use fo pactical flow analysis. Fluids ae composed of billions of molecules. Inteaction between molecules had to descibe/model. Howeve, useful fo specialized applications Spays, paticles, bubble dynamics, aefied gases. Coupled Euleian-Lagangian methods. Named afte Italian mathematician Joseph Louis Lagange ( ). 4-1

3 4-1 Lagangian and Euleian Desciptions (2) Euleian desciption of fluid flow: a flow domain o contol volume is defined by which fluid flows in and out. We define field vaiables which ae functions of space and time. Pessue field, P=P(x,y,z,t) Velocity field, V = V( x, y, zt, ) V= uxyzti+ vxyzt j+ wxyztk (,,, ) (,,, ) (,,, ) Acceleation field, a = a( x, y, z, t) a= a xyzti+ a xyzt j+ a xyztk (,,, ) (,,, ) (,,, ) x y z These (and othe) field vaiables define the flow field. Well suited fo fomulation of initial bounday-value poblems (PDE's). Named afte Swiss mathematician Leonhad Eule ( ). 1783) 4-2

4 4-1 Lagangian and Euleian Desciptions (3) Acceleation Field Conside a fluid paticle and Newton's second law, F = m a paticle paticle paticle The acceleation of the paticle is the time deivative of the paticle's velocity. dv paticle apaticle = dt Howeve, paticle velocity at a point is the same as the fluid velocity, V = V x ( t), y ( t), z ( t) ( ) paticle paticle paticle paticle To take the time deivative of, chain ule must be used. Vdt Vdx Vdy Vdz apaticle = t dt x dt y dt z dt paticle paticle paticle 4-3

5 4-1 Lagangian and Euleian Desciptions (4) dxpaticle dy paticle dz paticle Since u, v, w dt = dt = dt = V V V V apaticle = + u + v + w t x y z In vecto fom, the acceleation can be witten as dv V a( x, y, z, t) = = + ( V ) V dt t Fist tem is called the local acceleation and is nonzeo only fo unsteady flows. Second tem is called the advective acceleation and accounts fo the effect of the fluid paticle moving to a new location in the flow, whee the velocity is diffeent. 4-4

6 4-1 Lagangian and Euleian Desciptions (5) The total deivative opeato d/dt is call the mateial deivative and is often given special notation, D/Dt. DV dv V = = Dt dt t + ( V ) Advective acceleation is nonlinea: souce of many phenomenon and pimay challenge in solving fluid idflow poblems. Povides ``tansfomation'' between Lagangian and Euleian fames. Othe names fo the mateial deivative include: total,,p paticle, Lagangian, Euleian, and substantial deivative. V v t Steady: = 0 V 4-5

7 4-2 Fundamentals of Flow Visualization (1) Flow visualization is the visual examination of flowfield featues. Impotant fo both physical expeiments and numeical (CFD) solutions. Numeous methods Steamlines and steamtubes Pathlines Steaklines Timelines Refactive techniques Suface flow techniques 4-6

8 4-2 Fundamentals of Flow Visualization (2) A Steamline is a cuve that is eveywhee tangent to the instantaneous local velocity vecto. Conside an ac length d = dxi + dyj + dzk d must be paallel to the local velocity vecto V = ui + vj + wk Geometic aguments esults in the equation fo a steamline d dx dy dz = = = V u v w 4-7

9 4-2 Fundamentals of Flow Visualization (3) NASCAR suface pessue contous and steamlines Aiplane suface pessue contous, volume steamlines, and suface steamlines 4-8

10 4-2 Fundamentals of Flow Visualization (4) dx dt dy dt dz dt paticle paticle paticle = u( x, y, z, t) = υ( x, y, z, t) = w( x, y, z, t) A Pathline is the actual path taveled by an individual fluid paticle ove some time peiod. Same as the fluid paticle's mateial position vecto ( xpaticle ( t), ypaticle ( t), zpaticle ( t) ) Paticle location at time t: t = stat + x x Vdt t stat Paticle Image Velocimety (PIV) is a moden expeimental technique to measue velocity field ove a plane in the flow field. 4-9

11 4-2 Fundamentals of Flow Visualization (5) 4-10

12 4-2 Fundamentals of Flow Visualization (6) A Steakline is the locus of fluid paticles that have passed sequentially though a pescibed point in the flow. Easy to geneate in expeiments: dye in a wate flow, o smoke in an aiflow. 4-11

13 4-2 Fundamentals of Flow Visualization (7) 4-12

14 4-2 Fundamentals of Flow Visualization (8) Compaison Fo steady flow, steamlines, pathlines, and steaklines ae identical. Fo unsteady flow, they can be vey diffeent. Steamlines ae an instantaneous t pictue of the flow field Pathlines and Steaklines ae flow pattens that have a time histoy associated with them. Steakline: instantaneous snapshot of a time-integated flow patten. Pathline: time-exposed flow path of an individual paticle. 4-13

15 4-2 Fundamentals of Flow Visualization (9) EXAMPLE2.1 Steamlines and Pathlines in Two- Dimensional Flow A velocity field is given by ; the units of velocity ae m/s; x and y ae given in metes;. (a) Obtain an equation fo the steamlines in the xy plane. (b) Plot the Steamlines passing though the point (c) Detemine the velocity of paticle at the point (2, 8). (d) () If the paticle passing though the point ( is maked at time t = 0, detemine the location of the paticle at time t = 6 s. (e) What is the velocity of this paticle at time t = 6 s? (f) Show that the equation of the paticle path (the pathline) is the same as the equation of the Steamline. 4-14

16 4-2 Fundamentals of Flow Visualization (10) 4-15

17 4-3 Kinematic Desciption (1) In fluid mechanics, an element may undego fou fundamental types of motion. a) Tanslation b) Rotation c) Linea stain d) Shea stain Because fluids ae in constant motion, motion and defomation is best descibed in tems of ates a) velocity: ate of tanslation b) angula velocity: ate of otation c) linea stain ate: ate of linea stain d) shea stain ate: ate of shea stain 4-16

18 4-3 Kinematic Desciption (2) To be useful, these ates must be expessed in tems of velocity and deivatives of velocity The ate of tanslation vecto is descibed as the velocity vecto. In Catesian coodinates: V = ui + vj + wk Rate of otation at a point is defined as the aveage otation ate of two initially i i pependicula lines that intesect at that point. The ate of otation vecto in Catesian coodinates: 1 w v 1 u w 1 v u ω = i + j + k 2 y z 2 z x 2 x y 4-17

19 4-3 Kinematic Desciption (3) Linea Stain Rate is defined as the ate of incease in length pe unit length. In Catesian coodinates u xx, v ε = ε, ε w yy = zz = x y z Volumetic stain ate in Catesian coodinates 1 DV u v = ε w xx + εyy + εzz = + + V Dt x y z Since the volume of a fluid element is constant fo an incompessible flow, the volumetic stain ate must be zeo. 4-18

20 4-3 Kinematic Desciption (4) Shea Stain Rate at a point is defined as half of the ate of decease of the angle between two initially pependicula p lines that intesect at a point. Shea stain ate can be expessed in Catesian coodinates as: 1 u v 1 w u 1 v w εxy =, εzx, εyz 2 + = + = + x 2 x z 2 z y y 4-19

21 4-3 Kinematic Desciption (5) We can combine linea stain ate and shea stain ate into one symmetic second-ode tenso called the stain-ate tenso. u 1 u v 1 u w + + x 2 y x 2 z x εxx εxy ε xz 1 v u v 1 v w ε ij = εyx εyy εyz = 2 + x y y 2 + z y εzx εzy ε zz 1 w u 1 w v w x z 2 y z z 4-20

22 4-3 Kinematic Desciption (6) Pupose of ou discussion of fluid element kinematics: Bette appeciation of the inheent complexity of fluid dynamics Mathematical sophistication equied to fully descibe fluid motion Stain-ate ate tenso is impotant fo numeous easons. Fo example, Develop elationships between fluid stess and stain ate. Featue extaction and flow visualization in CFD simulations. 4-21

23 4-3 Kinematic Desciption (7) The voticity vecto is defined as the cul of the velocity vecto ζ = V Voticity is equal to twice the angula velocity of a fluid paticle. ζ = 2 ω Catesian coodinates w v u w v u ζ = i j k y z + z x + x y Cylindical coodinates ( u ) 1 u z uθ u uz θ u ζ = e + eθ + e θ z z θ In egions whee z = 0, the flow is called iotational. Elsewhee, the flow is called otational. z 4-22

24 4-3 Kinematic Desciption (8) 4-23

25 4-3 Kinematic Desciption (9) Special case: conside two flows with cicula steamlines u = 0, u = ω θ 2 ( u ) u ( ω ) 1 1 θ ζ = e = 0 e = 2ωe θ z z z K u = 0, uθ = 1 uθ u 1 K ζ = e = 0 e = 0e θ ( ) ( ) z z z 4-24

26 4-4 Reynolds Tanspot Theoem (RTT) (1) A system is a quantity of matte of fixed identity. No mass can coss a system bounday. A contol volume is a egion in space chosen fo study. Mass can coss a contol suface. The fundamental consevation laws (consevation of mass, enegy, and momentum) apply diectly to systems. Howeve, in most fluid mechanics poblems, contol volume analysis is pefeed ove system analysis (fo the same eason that the Euleian desciption is usually pefeed ove the Lagangian desciption). Theefoe, we need to tansfom the consevation laws fom a system to a contol volume. This is accomplished with the Reynolds tanspot theoem (RTT). 4-25

27 4-4 Reynolds Tanspot Theoem (RTT) (2) Let B epesent any extensive popety (such as mass, enegy, o momentum), and let b=b/m epesent the coesponding intensive i popety. Noting that t extensive popeties ae additive, the extensive popety B of the system at times t and t t can be expessed as 4-26

28 4-4 Reynolds Tanspot Theoem (RTT) (3) 4-27

29 4-4 Reynolds Tanspot Theoem (RTT) (4) 4-28

30 4-4 Reynolds Tanspot Theoem (RTT) (5) 4-29

31 4-4 Reynolds Tanspot Theoem (RTT) (6) Intepetation of the RTT: Time ate of change of the popety B of the system is equal to (Tem 1) + (Tem 2) Tem 1: the time ate of change of B of the contol volume Tem 2: the net flux of B out of the contol volume by mass cossing the contol suface db sys ( ρ ) dt = b dv + CV t CS ρ bv nda 4-30

32 4-4 Reynolds Tanspot Theoem (RTT) (7) Special Case 1: Steady Flow Steady flow : db dt sys = ρ b ( V n ) da CS Special Case 2: One-Dimensional Flow One - dimensional flow : db dt sys = d dt ρb dv + ρebev e Ae CV ρibv i i A i 1 out in fo each exit fo each exit db dt sys d = ρb dv + me be - mi bi dt CV out in 4-31

33 4-4 Reynolds Tanspot Theoem (RTT) (8) Mateial deivative (diffeential analysis): Db Dt b = + V t ( ) Geneal RTT, nonfixed CV (integal analysis): db dt sys = ( ρb) dv + CV t b CS ρbv nda Mass Momentum Enegy Angula momentum B, Extensive popeties m mv E V H b, Intensive popeties 1 e V V ( ) In Chaps 5 and 6, we will apply RTT to consevation of mass, enegy, linea momentum, and angula momentum. 4-32

34 4-4 Reynolds Tanspot Theoem (RTT) (9) 4-33

35 4-4 Reynolds Tanspot Theoem (RTT) (10) 4-34

36 4-4 Reynolds Tanspot Theoem (RTT) (11) 4-35

37 4-4 Reynolds Tanspot Theoem (RTT) (12) Thee is a diect analogy between the tansfomation fom Lagangian to Euleian desciptions (fo diffeential analysis using infinitesimally i i small fluid elements) and the tansfomation fom systems to contol volumes (fo integal analysis using lage, finite flow fields). 4-36

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