# Chapter 4: Fluid Kinematics

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2 Oveview Fluid kinematics deals with the motion of fluids without consideing the foces and moments which ceate the motion. Items discussed in this Chapte. Mateial deivative and its elationship to Lagangian and Euleian desciptions of fluid flow. Fundamental kinematic popeties of fluid motion and defomation. Reynolds Tanspot Theoem. Meccanica dei Fluidi I (ME) 2

3 Lagangian Desciption Lagangian desciption of fluid flow tacks the position and velocity of individual 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 ( ). Meccanica dei Fluidi I (ME) 3

4 Euleian Desciption 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, z, t) V= uxyzti+ vxyzt j+ wxyztk (,,, ) (,,, ) (,,, ) Acceleation field, a = a( x, y, z, t) a= a xyzt i+ a xyzt j+ a xyzt k (,,, ) (,,, ) (,,, ) 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 ( ). Meccanica dei Fluidi I (ME) 4

5 Example: Coupled Euleian-Lagangian Method Foensic analysis of Columbia accident: simulation of shuttle debis tajectoy using Euleian CFD fo flow field and Lagangian method fo the debis. Meccanica dei Fluidi I (ME) 5

6 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: dvpaticle 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, chain ule must be used. Vdt Vdxpaticle Vdy paticle Vdz paticle apaticle = t dt x dt y dt z dt, t) Meccanica dei Fluidi I (ME) 6

7 Acceleation Field Since dxpaticle dy paticle dz paticle = 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 (o convective) acceleation and accounts fo the effect of the fluid paticle moving to a new location in the flow, whee the velocity is diffeent (it can thus be nonzeo even fo steady flows). Meccanica dei Fluidi I (ME) 7

8 Mateial Deivative 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. ) V Advective acceleation is nonlinea: souce of many phenomena and pimay challenge in solving fluid flow poblems. Povides tansfomation ' between Lagangian and Euleian fames. Othe names fo the mateial deivative include: total, paticle, Lagangian, Euleian, and substantial deivative. Meccanica dei Fluidi I (ME) 8

9 Flow Visualization Flow visualization is the visual examination of flow-field featues. Impotant fo both physical expeiments and numeical (CFD) solutions. Numeous methods Steamlines and steamtubes Pathlines Steaklines Timelines Refactive flow visualization techniques Suface flow visualization techniques Meccanica dei Fluidi I (ME) 9

10 Steamlines and steamtubes A steamline is a cuve that is eveywhee tangent to the instantaneous local velocity vecto. Conside an infinitesimal ac length along a steamline: d = dxi + dyj + dzk d By definition must be paallel to the local velocity vecto V = ui + vj + wk Geometic aguments esult in the equation fo a steamline d dx dy dz = = = V u v w Meccanica dei Fluidi I (ME) 10

11 Steamlines and steamtubes NASCAR suface pessue contous and steamlines Aiplane suface pessue contous, volume steamlines, and suface steamlines Meccanica dei Fluidi I (ME) 11

12 Steamlines and steamtubes A steamtube consists of a bundle of individual steamlines. Since fluid cannot coss a steamline (by definition), fluid within a steamtube must emain thee. Steamtubes ae, obviously, instantaneous quantities and they may change significantly with time. In the conveging potion of an incompessible flow field, the diamete of the steamtube must decease as the velocity inceases, so as to conseve mass. Meccanica dei Fluidi I (ME) 12

13 Pathlines 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 + t x x Vdt stat Paticle Image Velocimety (PIV) is a moden expeimental technique to measue velocity field ove a plane in the flow field. Meccanica dei Fluidi I (ME) 13

14 Steaklines A steakline is the locus of fluid paticles that have passed sequentially though a pescibed point in the flow. Easy to geneate in expeiments: continuous intoduction of dye (in a wate flow) o smoke (in an aiflow) fom a point. Meccanica dei Fluidi I (ME) 14

15 Compaisons If the flow is steady, steamlines, pathlines and steaklines ae identical. Fo unsteady flows, they can be vey diffeent. Steamlines povide an instantaneous pictue of the flow field Pathlines and steaklines ae flow pattens that have a time histoy associated with them. Meccanica dei Fluidi I (ME) 15

16 Timelines A timeline is a set of adjacent fluid paticles that wee maked at the same (ealie) instant in time. Expeimentally, timelines can be geneated using a hydogen bubble wie: a line is maked and its movement/defomation is followed in time. Meccanica dei Fluidi I (ME) 16

17 Plots of Data A Pofile plot indicates how the value of a scala popety vaies along some desied diection in the flow field. A Vecto plot is an aay of aows indicating the magnitude and diection of a vecto popety at an instant in time. A Contou plot shows cuves of constant values of a scala popety (o magnitude fo a vecto popety) at an instant in time. Meccanica dei Fluidi I (ME) 17

18 Kinematic Desciption In fluid mechanics (as in solid 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 Meccanica dei Fluidi I (ME) 18

19 Rate of Tanslation and Rotation To be useful, these defomation ates must be expessed in tems of velocity and deivatives of velocity The ate of tanslation vecto is descibed mathematically as the velocity vecto. In Catesian coodinates: V = ui + vj + wk Rate of otation (angula velocity) at a point is defined as the aveage otation ate of two lines which ae initially pependicula and that intesect at that point. Meccanica dei Fluidi I (ME) 19

20 Rate of Rotation In 2D the aveage otation angle of the fluid element about the point P is α ω = (α a + α b )/2 The ate of otation of the fluid element about P is 1 u w 1 v u i + ω j= + k 2 z x 2 x y In 3D the angula velocity vecto is: 1 w v 1 u w 1 v u ω = i + j + k 2 y z 2 z x 2 x y Meccanica dei Fluidi I (ME) 20

21 Linea Stain Rate Linea Stain Rate is defined as the ate of incease in length pe unit length. In Catesian coodinates u xx, v ε = εyy =, ε w zz = x y z The ate of incease of volume of a fluid element pe unit volume is the volumetic stain ate, in Catesian coodinates: 1 DV u v = ε w xx + εyy + εzz = + + V Dt x y z (we ae talking about a mateial volume, hence the D) Since the volume of a fluid element is constant fo an incompessible flow, the volumetic stain ate must be zeo. Meccanica dei Fluidi I (ME) 21

22 Shea Stain Rate Shea Stain Rate at a point is defined as half the ate of decease of the angle between two initially pependicula lines that intesect at a point. positive shea stain negative shea stain Meccanica dei Fluidi I (ME) 22

23 Shea Stain Rate The shea stain at point P 1 is ε xy = - d α dt a-b 2 Shea stain ate can be expessed in Catesian coodinates as: 1 u v 1 1 xy, w u zx, v ε ε ε w = yz 2 + = + = + y x 2 x z 2 z y Meccanica dei Fluidi I (ME) 23

24 Shea Stain Rate We can combine linea stain ate and shea stain ate into one symmetic second-ode tenso called E: stain-ate tenso. In Catesian coodinates: 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 εij = 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 E Meccanica dei Fluidi I (ME) 24

25 State of Motion Paticle moves fom O to P in time t Taylo seies aound O (fo a small displacement) yields: The tenso u can be split into a symmetic pat (E, the stain tenso) and an antisymmetic pat (Ω, the otation tenso) pat as so that Meccanica dei Fluidi I (ME) 25

26 Shea Stain Rate 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 tenso is impotant fo numeous easons. Fo example, Develop elationships between fluid stess and stain ate. Featue extaction and flow visualization in CFD simulations. Meccanica dei Fluidi I (ME) 26

27 Tanslation, Rotation, Linea Stain, Shea Stain, and Volumetic Stain Defomation of fluid elements (made visible with a tace) duing thei compessible motion though a convegent channel; shea stain is moe evident nea the walls because of lage velocity gadients (a bounday laye is pesent thee). Meccanica dei Fluidi I (ME) 27

28 Stain Rate Tenso Example: Visualization of tailing-edge tubulent eddies fo a hydofoil with a beveled tailing edge Featue extaction method is based upon eigen-analysis of the stain-ate tenso. Meccanica dei Fluidi I (ME) 28

29 Voticity and Rotationality 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 = 2ω Catesian coodinates w v u w v u ζ = i + j + k y z z x x y Cylindical coodinates ( u ) 1 uz uθ u uz θ u ζ = e + eθ + e θ z z θ In egions whee ζ = 0, the flow is called iotational. Elsewhee, the flow is called otational. z Meccanica dei Fluidi I (ME) 29

30 Voticity and Rotationality Meccanica dei Fluidi I (ME) 30

31 Compaison of Two Cicula Flows Special case: conside two flows with cicula steamlines u = 0, u = ω θ solid-body otation 2 ( u ) u ( ω ) 1 θ 1 ζ = e = 0 e = 2ωe θ z z z K line votex u = 0, uθ = 1 ( uθ ) u 1 ( K ) ζ = e = 0 e = 0e θ z z z Meccanica dei Fluidi I (ME) 31

32 Reynolds Tanspot Theoem (RTT) 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). Meccanica dei Fluidi I (ME) 32

33 Reynolds Tanspot Theoem (RTT) Thee is a diect analogy between the tansfomation fom Lagangian to Euleian desciptions (fo diffeential analysis using infinitesimally small fluid elements) and the tansfomation fom systems to contol volumes (fo integal analysis using lage, finite flow fields). Meccanica dei Fluidi I (ME) 33

34 Reynolds Tanspot Theoem (RTT) Mateial deivative (diffeential analysis): Db Dt b = +. t ( V ) RTT, moving o defomable CV (integal analysis): b V = V - V cs Mass Momentum Enegy Angula momentum B, Extensive popeties m mv E H b, Intensive popeties 1 V e ( V) In Chaps 5 and 6, we will apply RTT to consevation of mass, enegy, linea momentum, and angula momentum. Meccanica dei Fluidi I (ME) 34

35 Reynolds Tanspot Theoem (RTT) RTT, fixed CV: db dt sys = ( ρbdv ) V + CV t CS ρbvnda. Time ate of change of the popety B of the closed system is equal to (Tem 1) + (Tem 2) Tem 1: time ate of change of B of the contol volume Tem 2: net flux of B out of the contol volume by mass cossing the contol suface Meccanica dei Fluidi I (ME) 35

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