# Chapter 4: Fluid Kinematics

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2 Overview Fluid kinematics deals with the motion of fluids without considering the forces and moments which create the motion. Items discussed in this Chapter. Material derivative and its relationship to Lagrangian and Eulerian descriptions of fluid flow. Fundamental kinematic properties of fluid motion and deformation. Reynolds Transport Theorem. Meccanica dei Fluidi I 2

3 Lagrangian Description Lagrangian description of fluid flow tracks the position and velocity of individual particles. Based upon Newton's laws of motion. Difficult to use for practical flow analysis. Fluids are composed of billions of molecules. Interaction between molecules hard to describe/model. However, useful for specialized applications Sprays, particles, bubble dynamics, rarefied gases. Coupled Eulerian-Lagrangian methods. Named after Italian mathematician Joseph Louis Lagrange ( ). Meccanica dei Fluidi I 3

4 Eulerian Description Eulerian description of fluid flow: a flow domain or control volume is defined by which fluid flows in and out. We define field variables which are functions of space and time. Pressure field, P = P(x,y,z,t) Velocity field, V V x, y, z, t Acceleration field,,,,,,,,,, V u x y z t i v x y z t j w x y z t k a a x, y, z, t,,,,,,,,, a a x y z t i a x y z t j a x y z t k x y z These (and other) field variables define the flow field. Well suited for formulation of initial boundary-value problems (PDE's). Named after Swiss mathematician Leonhard Euler ( ). Meccanica dei Fluidi I 4

5 Example: Coupled Eulerian-Lagrangian Method Forensic analysis of Columbia accident: simulation of shuttle debris trajectory using Eulerian CFD for flow field and Lagrangian method for the debris. Meccanica dei Fluidi I 5

6 Acceleration Field Consider a fluid particle and Newton's second law, Fparticle mparticlea particle The acceleration of the particle is the time derivative of the particle's velocity: dvparticle aparticle dt However, particle velocity at a point is the same as the fluid velocity, V V x t, y t, z t particle particle particle particle To take the time derivative, chain rule must be used. a particle V dt V dx V dy V dz t dt x dt y dt z dt particle particle particle, t) Meccanica dei Fluidi I 6

7 Acceleration Field Since dxparticle dy particle dz particle u, v, w dt dt dt V V V V aparticle u v w t x y z In vector form, the acceleration can be written as dv V ax, y, z, t V. V dt t First term is called the local acceleration and is nonzero only for unsteady flows. Second term is called the advective (or convective) acceleration and accounts for the effect of the fluid particle moving to a new location in the flow, where the velocity is different (it can thus be nonzero even for steady flows). Meccanica dei Fluidi I 7

8 Material Derivative The total derivative operator d/dt is call the material derivative and is often given special notation, D/Dt. DV dv V V. Dt dt t Advective acceleration is nonlinear: source of many phenomena and primary challenge in solving fluid flow problems. Provides transformation ' between Lagrangian and Eulerian frames. Other names for the material derivative include: total, particle, Lagrangian, Eulerian, and substantial derivative. V Meccanica dei Fluidi I 8

9 Flow Visualization Flow visualization is the visual examination of flow-field features. Important for both physical experiments and numerical (CFD) solutions. Numerous methods Streamlines and streamtubes Pathlines Streaklines Timelines Refractive flow visualization techniques Surface flow visualization techniques Meccanica dei Fluidi I 9

10 Streamlines and streamtubes A streamline is a curve that is everywhere tangent to the instantaneous local velocity vector. Consider an infinitesimal arc length along a streamline: dr dxi dyj dzk dr By definition must be parallel to the local velocity vector V ui vj wk Geometric arguments result in the equation for a streamline dr dx dy dz V u v w Meccanica dei Fluidi I 10

11 Streamlines and streamtubes NASCAR surface pressure contours and streamlines Airplane surface pressure contours, volume streamlines, and surface streamlines Meccanica dei Fluidi I 11

12 Streamlines and streamtubes A streamtube consists of a bundle of individual streamlines. Since fluid cannot cross a streamline (by definition), fluid within a streamtube must remain there. Streamtubes are, obviously, instantaneous quantities and they may change significantly with time. In the converging portion of an incompressible flow field, the diameter of the streamtube must decrease as the velocity increases, so as to conserve mass. Meccanica dei Fluidi I 12

13 Pathlines A pathline is the actual path traveled by an individual fluid particle over some time period. Same as the fluid particle's material position vector,, x t y t z t particle particle particle Particle location at time t: x xstart Vdt start Particle Image Velocimetry (PIV) is a modern experimental technique to measure velocity field over a plane in the flow field. t t Meccanica dei Fluidi I 13

14 Streaklines A streakline is the locus of fluid particles that have passed sequentially through a prescribed point in the flow. Easy to generate in experiments: continuous introduction of dye (in a water flow) or smoke (in an airflow) from a point. Meccanica dei Fluidi I 14

15 Comparisons If the flow is steady, streamlines, pathlines and streaklines are identical. For unsteady flows, they can be very different. Streamlines provide an instantaneous picture of the flow field Pathlines and streaklines are flow patterns that have a time history associated with them. Meccanica dei Fluidi I 15

16 Timelines A timeline is a set of adjacent fluid particles that were marked at the same (earlier) instant in time. Experimentally, timelines can be generated using a hydrogen bubble wire: a line is marked and its movement/deformation is followed in time. Meccanica dei Fluidi I 16

17 Plots of Data A Profile plot indicates how the value of a scalar property varies along some desired direction in the flow field. A Vector plot is an array of arrows indicating the magnitude and direction of a vector property at an instant in time. A Contour plot shows curves of constant values of a scalar property (or magnitude for a vector property) at an instant in time. Meccanica dei Fluidi I 17

18 Kinematic Description In fluid mechanics (as in solid mechanics), an element may undergo four fundamental types of motion. a) Translation b) Rotation c) Linear strain d) Shear strain Because fluids are in constant motion, motion and deformation is best described in terms of rates a) velocity: rate of translation b) angular velocity: rate of rotation c) linear strain rate: rate of linear strain d) shear strain rate: rate of shear strain Meccanica dei Fluidi I 18

19 Rate of Translation and Rotation To be useful, these deformation rates must be expressed in terms of velocity and derivatives of velocity The rate of translation vector is described mathematically as the velocity vector. In Cartesian coordinates: V ui vj wk Rate of rotation (angular velocity) at a point is defined as the average rotation rate of two lines which are initially perpendicular and that intersect at that point. Meccanica dei Fluidi I 19

20 Rate of Rotation In 2D the average rotation angle of the fluid element about the point P is a w = (a a + a b )/2 The rate of rotation of the fluid element about P is 1 u w 1 v u i j k 2 z x 2 x y w In 3D the angular velocity vector is: 1 w v 1 u w 1 v u w i j k 2 y z 2 z x 2 x y Meccanica dei Fluidi I 20

21 Linear Strain Rate Linear Strain Rate is defined as the rate of increase in length per unit length. In Cartesian coordinates u xx, v yy, w zz x y z The rate of increase of volume of a fluid element per unit volume is the volumetric strain rate, in Cartesian coordinates: 1 DV u v w xx yy zz V Dt x y z (we are talking about a material volume, hence the D) Since the volume of a fluid element is constant for an incompressible flow, the volumetric strain rate must be zero. Meccanica dei Fluidi I 21

22 Shear Strain Rate Shear Strain Rate at a point is defined as half the rate of decrease of the angle between two initially perpendicular lines that intersect at a point. positive shear strain negative shear strain Meccanica dei Fluidi I 22

23 Shear Strain Rate The shear strain at point P 1 is xy = - d aa-b dt 2 Shear strain rate can be expressed in Cartesian coordinates as: 1 u v 1 1 xy, w u zx, v w yz 2 y x 2 x z 2 z y Meccanica dei Fluidi I 23

24 Shear Strain Rate We can combine linear strain rate and shear strain rate into one symmetric second-order tensor called E: strain-rate tensor. In Cartesian coordinates: u 1u v 1u w x 2 y x 2z x xx xy xz 1 v u v 1 v w ij yx yy yz 2 x y y 2 z y zx zy zz 1w u 1w v w 2 x z 2 y z z E Meccanica dei Fluidi I 24

25 State of Motion Particle moves from O to P in time D t Taylor series around O (for a small displacement) yields: The tensor u can be split into a symmetric part (E, the strain tensor) and an antisymmetric part W, the rotation tensor part as so that Meccanica dei Fluidi I 25

26 Shear Strain Rate Purpose of our discussion of fluid element kinematics: Better appreciation of the inherent complexity of fluid dynamics Mathematical sophistication required to fully describe fluid motion Strain-rate tensor is important for numerous reasons. For example, Develop relationships between fluid stress and strain rate. Feature extraction and flow visualization in CFD simulations. Meccanica dei Fluidi I 26

27 Translation, Rotation, Linear Strain, Shear Strain, and Volumetric Strain Deformation of fluid elements (made visible with a tracer) during their compressible motion through a convergent channel; shear strain is more evident near the walls because of larger velocity gradients (a boundary layer is present there). Meccanica dei Fluidi I 27

28 Strain Rate Tensor Example: Visualization of trailing-edge turbulent eddies for a hydrofoil with a beveled trailing edge Feature extraction method is based upon eigen-analysis of the strain-rate tensor. Meccanica dei Fluidi I 28

29 Vorticity and Rotationality The vorticity vector is defined as the curl of the velocity vector z V Vorticity is equal to twice the angular velocity of a fluid particle: z 2w Cartesian coordinates w v u w v u z i j k y z z x x y Cylindrical coordinates 1 uz u ru ur uz u r z er e e r z z r r In regions where z = 0, the flow is called irrotational. Elsewhere, the flow is called rotational. z Meccanica dei Fluidi I 29

30 Vorticity and Rotationality Meccanica dei Fluidi I 30

31 Comparison of Two Circular Flows Special case: consider two flows with circular streamlines u 0, u wr r solid-body rotation 2 ru u wr 1 r 1 z e 0 e 2we r r r r z z z K line vortex ur 0, u r 1ru u r 1K z e 0e 0e r r r r z z z Meccanica dei Fluidi I 31

32 Circulation and vorticity Circulation: Meccanica dei Fluidi I 32

33 Circulation and vorticity Stokes theorem: If the flow is irrotational everywhere within the contour of integration C then G = 0. Meccanica dei Fluidi I 33

34 Reynolds Transport Theorem (RTT) A system is a quantity of matter of fixed identity. No mass can cross a system boundary. A control volume is a region in space chosen for study. Mass can cross a control surface. The fundamental conservation laws (conservation of mass, energy, and momentum) apply directly to systems. However, in most fluid mechanics problems, control volume analysis is preferred over system analysis (for the same reason that the Eulerian description is usually preferred over the Lagrangian description). Therefore, we need to transform the conservation laws from a system to a control volume. This is accomplished with the Reynolds transport theorem (RTT). Meccanica dei Fluidi I 34

35 Reynolds Transport Theorem (RTT) There is a direct analogy between the transformation from Lagrangian to Eulerian descriptions (for differential analysis using infinitesimally small fluid elements) and the transformation from systems to control volumes (for integral analysis using large, finite flow fields). Meccanica dei Fluidi I 35

36 Reynolds Transport Theorem (RTT) Material derivative (differential analysis): Db Dt b V. t RTT, moving or deformable CV (integral analysis): b V r = V - V cs Mass Momentum Energy Angular momentum B, Extensive properties m mv E b, Intensive properties 1 V e In Chaps 5 and 6, we will apply RTT to conservation of mass, energy, linear momentum, and angular momentum. r H V Meccanica dei Fluidi I 36

37 Reynolds Transport Theorem (RTT) RTT, fixed CV: db dt sys CV t b dvv CS. bv nda Time rate of change of the property B of the closed system is equal to (Term 1) + (Term 2) Term 1: time rate of change of B of the control volume Term 2: net flux of B out of the control volume by mass crossing the control surface Meccanica dei Fluidi I 37

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